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1.1: Statements and Conditional Statements

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• Page ID 7034

• Ted Sundstrom
• Grand Valley State University via ScholarWorks @Grand Valley State University

Much of our work in mathematics deals with statements. In mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition . The key is that there must be no ambiguity. To be a statement, a sentence must be true or false, and it cannot be both. So a sentence such as "The sky is beautiful" is not a statement since whether the sentence is true or not is a matter of opinion. A question such as "Is it raining?" is not a statement because it is a question and is not declaring or asserting that something is true.

Some sentences that are mathematical in nature often are not statements because we may not know precisely what a variable represents. For example, the equation 2\(x\)+5 = 10 is not a statement since we do not know what \(x\) represents. If we substitute a specific value for \(x\) (such as \(x\) = 3), then the resulting equation, 2\(\cdot\)3 +5 = 10 is a statement (which is a false statement). Following are some more examples:

• There exists a real number \(x\) such that 2\(x\)+5 = 10. This is a statement because either such a real number exists or such a real number does not exist. In this case, this is a true statement since such a real number does exist, namely \(x\) = 2.5.
• For each real number \(x\), \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\). This is a statement since either the sentence \(2x +5 = 2 \left( x + \dfrac{5}{2}\right)\) is true when any real number is substituted for \(x\) (in which case, the statement is true) or there is at least one real number that can be substituted for \(x\) and produce a false statement (in which case, the statement is false). In this case, the given statement is true.
• Solve the equation \(x^2 - 7x +10 =0\). This is not a statement since it is a directive. It does not assert that something is true.
• \((a+b)^2 = a^2+b^2\) is not a statement since it is not known what \(a\) and \(b\) represent. However, the sentence, “There exist real numbers \(a\) and \(b\) such that \((a+b)^2 = a^2+b^2\)" is a statement. In fact, this is a true statement since there are such integers. For example, if \(a=1\) and \(b=0\), then \((a+b)^2 = a^2+b^2\).
• Compare the statement in the previous item to the statement, “For all real numbers \(a\) and \(b\), \((a+b)^2 = a^2+b^2\)." This is a false statement since there are values for \(a\) and \(b\) for which \((a+b)^2 \ne a^2+b^2\). For example, if \(a=2\) and \(b=3\), then \((a+b)^2 = 5^2 = 25\) and \(a^2 + b^2 = 2^2 +3^2 = 13\).

Progress Check 1.1: Statements

Which of the following sentences are statements? Do not worry about determining whether a statement is true or false; just determine whether each sentence is a statement or not.

• 2\(\cdot\)7 + 8 = 22.
• \((x-1) = \sqrt(x + 11)\).
• \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(2x + 5y = 7\).
• There are integers \(x\) and \(y\) such that \(23x + 27y = 52\).
• Given a line \(L\) and a point \(P\) not on that line, there is a unique line through \(P\) that does not intersect \(L\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).
• \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) for all real numbers \(a\) and \(b\).
• The derivative of \(f(x) = \sin x\) is \(f' (x) = \cos x\).
• Does the equation \(3x^2 - 5x - 7 = 0\) have two real number solutions?
• If \(ABC\) is a right triangle with right angle at vertex \(B\), and if \(D\) is the midpoint of the hypotenuse, then the line segment connecting vertex \(B\) to \(D\) is half the length of the hypotenuse.
• There do not exist three integers \(x\), \(y\), and \(z\) such that \(x^3 + y^2 = z^3\).

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How Do We Decide If a Statement Is True or False?

In mathematics, we often establish that a statement is true by writing a mathematical proof. To establish that a statement is false, we often find a so-called counterexample. (These ideas will be explored later in this chapter.) So mathematicians must be able to discover and construct proofs. In addition, once the discovery has been made, the mathematician must be able to communicate this discovery to others who speak the language of mathematics. We will be dealing with these ideas throughout the text.

For now, we want to focus on what happens before we start a proof. One thing that mathematicians often do is to make a conjecture beforehand as to whether the statement is true or false. This is often done through exploration. The role of exploration in mathematics is often difficult because the goal is not to find a specific answer but simply to investigate. Following are some techniques of exploration that might be helpful.

Techniques of Exploration

• Guesswork and conjectures . Formulate and write down questions and conjectures. When we make a guess in mathematics, we usually call it a conjecture.

For example, if someone makes the conjecture that \(\sin(2x) = 2 \sin(x)\), for all real numbers \(x\), we can test this conjecture by substituting specific values for \(x\). One way to do this is to choose values of \(x\) for which \(\sin(x)\)is known. Using \(x = \frac{\pi}{4}\), we see that

\(\sin(2(\frac{\pi}{4})) = \sin(\frac{\pi}{2}) = 1,\) and

\(2\sin(\frac{\pi}{4}) = 2(\frac{\sqrt2}{2}) = \sqrt2\).

Since \(1 \ne \sqrt2\), these calculations show that this conjecture is false. However, if we do not find a counterexample for a conjecture, we usually cannot claim the conjecture is true. The best we can say is that our examples indicate the conjecture is true. As an example, consider the conjecture that

If \(x\) and \(y\) are odd integers, then \(x + y\) is an even integer.

We can do lots of calculation, such as \(3 + 7 = 10\) and \(5 + 11 = 16\), and find that every time we add two odd integers, the sum is an even integer. However, it is not possible to test every pair of odd integers, and so we can only say that the conjecture appears to be true. (We will prove that this statement is true in the next section.)

• Use of prior knowledge. This also is very important. We cannot start from square one every time we explore a statement. We must make use of our acquired mathematical knowledge. For the conjecture that \(\sin (2x) = 2 \sin(x)\), for all real numbers \(x\), we might recall that there are trigonometric identities called “double angle identities.” We may even remember the correct identity for \(\sin (2x)\), but if we do not, we can always look it up. We should recall (or find) that for all real numbers \(x\), \[\sin(2x) = 2 \sin(x)\cos(x).\]
• We could use this identity to argue that the conjecture “for all real numbers \(x\), \(\sin (2x) = 2 \sin(x)\)” is false, but if we do, it is still a good idea to give a specific counterexample as we did before.
• Cooperation and brainstorming . Working together is often more fruitful than working alone. When we work with someone else, we can compare notes and articulate our ideas. Thinking out loud is often a useful brainstorming method that helps generate new ideas.

Progress Check 1.2: Explorations

Use the techniques of exploration to investigate each of the following statements. Can you make a conjecture as to whether the statement is true or false? Can you determine whether it is true or false?

• \((a + b)^2 = a^2 + b^2\), for all real numbers a and b.
• There are integers \(x\) and \(y\) such that \(2x + 5y = 41\).
• If \(x\) is an even integer, then \(x^2\) is an even integer.
• If \(x\) and \(y\) are odd integers, then \(x \cdot y\) is an odd integer.

Conditional Statements

One of the most frequently used types of statements in mathematics is the so-called conditional statement. Given statements \(P\) and \(Q\), a statement of the form “If \(P\) then \(Q\)” is called a conditional statement . It seems reasonable that the truth value (true or false) of the conditional statement “If \(P\) then \(Q\)” depends on the truth values of \(P\) and \(Q\). The statement “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. The statement \(P\) is called the hypothesis of the conditional statement, and the statement \(Q\) is called the conclusion of the conditional statement. Since conditional statements are probably the most important type of statement in mathematics, we give a more formal definition.

A conditional statement is a statement that can be written in the form “If \(P\) then \(Q\),” where \(P\) and \(Q\) are sentences. For this conditional statement, \(P\) is called the hypothesis and \(Q\) is called the conclusion .

Intuitively, “If \(P\) then \(Q\)” means that \(Q\) must be true whenever \(P\) is true. Because conditional statements are used so often, a symbolic shorthand notation is used to represent the conditional statement “If \(P\) then \(Q\).” We will use the notation \(P \to Q\) to represent “If \(P\) then \(Q\).” When \(P\) and \(Q\) are statements, it seems reasonable that the truth value (true or false) of the conditional statement \(P \to Q\) depends on the truth values of \(P\) and \(Q\). There are four cases to consider:

• \(P\) is true and \(Q\) is true.
• \(P\) is false and \(Q\) is true.
• \(P\) is true and \(Q\) is false.
• \(P\) is false and \(Q\) is false.

The conditional statement \(P \to Q\) means that \(Q\) is true whenever \(P\) is true. It says nothing about the truth value of \(Q\) when \(P\) is false. Using this as a guide, we define the conditional statement \(P \to Q\) to be false only when \(P\) is true and \(Q\) is false, that is, only when the hypothesis is true and the conclusion is false. In all other cases, \(P \to Q\) is true. This is summarized in Table 1.1 , which is called a truth table for the conditional statement \(P \to Q\). (In Table 1.1 , T stands for “true” and F stands for “false.”)

Table 1.1: Truth Table for \(P \to Q\)

The important thing to remember is that the conditional statement \(P \to Q\) has its own truth value. It is either true or false (and not both). Its truth value depends on the truth values for \(P\) and \(Q\), but some find it a bit puzzling that the conditional statement is considered to be true when the hypothesis P is false. We will provide a justification for this through the use of an example.

Example 1.3:

Suppose that I say

“If it is not raining, then Daisy is riding her bike.”

We can represent this conditional statement as \(P \to Q\) where \(P\) is the statement, “It is not raining” and \(Q\) is the statement, “Daisy is riding her bike.”

Although it is not a perfect analogy, think of the statement \(P \to Q\) as being false to mean that I lied and think of the statement \(P \to Q\) as being true to mean that I did not lie. We will now check the truth value of \(P \to Q\) based on the truth values of \(P\) and \(Q\).

• Suppose that both \(P\) and \(Q\) are true. That is, it is not raining and Daisy is riding her bike. In this case, it seems reasonable to say that I told the truth and that\(P \to Q\) is true.
• Suppose that \(P\) is true and \(Q\) is false or that it is not raining and Daisy is not riding her bike. It would appear that by making the statement, “If it is not raining, then Daisy is riding her bike,” that I have not told the truth. So in this case, the statement \(P \to Q\) is false.
• Now suppose that \(P\) is false and \(Q\) is true or that it is raining and Daisy is riding her bike. Did I make a false statement by stating that if it is not raining, then Daisy is riding her bike? The key is that I did not make any statement about what would happen if it was raining, and so I did not tell a lie. So we consider the conditional statement, “If it is not raining, then Daisy is riding her bike,” to be true in the case where it is raining and Daisy is riding her bike.
• Finally, suppose that both \(P\) and \(Q\) are false. That is, it is raining and Daisy is not riding her bike. As in the previous situation, since my statement was \(P \to Q\), I made no claim about what would happen if it was raining, and so I did not tell a lie. So the statement \(P \to Q\) cannot be false in this case and so we consider it to be true.

Progress Check 1.4: xplorations with Conditional Statements

1 . Consider the following sentence:

If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.

Although the hypothesis and conclusion of this conditional sentence are not statements, the conditional sentence itself can be considered to be a statement as long as we know what possible numbers may be used for the variable \(x\). From the context of this sentence, it seems that we can substitute any positive real number for \(x\). We can also substitute 0 for \(x\) or a negative real number for x provided that we are willing to work with a false hypothesis in the conditional statement. (In Chapter 2 , we will learn how to be more careful and precise with these types of conditional statements.)

(a) Notice that if \(x = -3\), then \(x^2 + 8x = -15\), which is negative. Does this mean that the given conditional statement is false?

(b) Notice that if \(x = 4\), then \(x^2 + 8x = 48\), which is positive. Does this mean that the given conditional statement is true?

(c) Do you think this conditional statement is true or false? Record the results for at least five different examples where the hypothesis of this conditional statement is true.

2 . “If \(n\) is a positive integer, then \(n^2 - n +41\) is a prime number.” (Remember that a prime number is a positive integer greater than 1 whose only positive factors are 1 and itself.) To explore whether or not this statement is true, try using (and recording your results) for \(n = 1\), \(n = 2\), \(n = 3\), \(n = 4\), \(n = 5\), and \(n = 10\). Then record the results for at least four other values of \(n\). Does this conditional statement appear to be true?

Further Remarks about Conditional Statements

Suppose that Ed has exactly $52 in his wallet. The following four statements will use the four possible truth combinations for the hypothesis and conclusion of a conditional statement.

• If Ed has exactly $52 in his wallet, then he has $20 in his wallet. This is a true statement. Notice that both the hypothesis and the conclusion are true.
• If Ed has exactly $52 in his wallet, then he has $100 in his wallet. This statement is false. Notice that the hypothesis is true and the conclusion is false.
• If Ed has $100 in his wallet, then he has at least $50 in his wallet. This statement is true regardless of how much money he has in his wallet. In this case, the hypothesis is false and the conclusion is true.

This is admittedly a contrived example but it does illustrate that the conventions for the truth value of a conditional statement make sense. The message is that in order to be complete in mathematics, we need to have conventions about when a conditional statement is true and when it is false.

If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number.

Perhaps for all of the values you tried for \(n\), \((n^2 - n + 41)\) turned out to be a prime number. However, if we try \(n = 41\), we ge \(n^2 - n + 41 = 41^2 - 41 + 41\) \(n^2 - n + 41 = 41^2\) So in the case where \(n = 41\), the hypothesis is true (41 is a positive integer) and the conclusion is false \(41^2\) is not prime. Therefore, 41 is a counterexample for this conjecture and the conditional statement “If \(n\) is a positive integer, then \((n^2 - n + 41)\) is a prime number” is false. There are other counterexamples (such as \(n = 42\), \(n = 45\), and \(n = 50\)), but only one counterexample is needed to prove that the statement is false.

• Although one example can be used to prove that a conditional statement is false, in most cases, we cannot use examples to prove that a conditional statement is true. For example, in Progress Check 1.4 , we substituted values for \(x\) for the conditional statement “If \(x\) is a positive real number, then \(x^2 + 8x\) is a positive real number.” For every positive real number used for \(x\), we saw that \(x^2 + 8x\) was positive. However, this does not prove the conditional statement to be true because it is impossible to substitute every positive real number for \(x\). So, although we may believe this statement is true, to be able to conclude it is true, we need to write a mathematical proof. Methods of proof will be discussed in Section 1.2 and Chapter 3 .

Progress Check 1.5: Working with a Conditional Statement

The following statement is a true statement, which is proven in many calculus texts.

If the function \(f\) is differentiable at \(a\), then the function \(f\) is continuous at \(a\).

Using only this true statement, is it possible to make a conclusion about the function in each of the following cases?

• It is known that the function \(f\), where \(f(x) = \sin x\), is differentiable at 0.
• It is known that the function \(f\), where \(f(x) = \sqrt[3]x\), is not differentiable at 0.
• It is known that the function \(f\), where \(f(x) = |x|\), is continuous at 0.
• It is known that the function \(f\), where \(f(x) = \dfrac{|x|}{x}\) is not continuous at 0.

Closure Properties of Number Systems

The primary number system used in algebra and calculus is the real number system . We usually use the symbol R to stand for the set of all real numbers. The real numbers consist of the rational numbers and the irrational numbers. The rational numbers are those real numbers that can be written as a quotient of two integers (with a nonzero denominator), and the irrational numbers are those real numbers that cannot be written as a quotient of two integers. That is, a rational number can be written in the form of a fraction, and an irrational number cannot be written in the form of a fraction. Some common irrational numbers are \(\sqrt2\), \(\pi\) and \(e\). We usually use the symbol \(\mathbb{Q}\) to represent the set of all rational numbers. (The letter \(\mathbb{Q}\) is used because rational numbers are quotients of integers.) There is no standard symbol for the set of all irrational numbers.

Perhaps the most basic number system used in mathematics is the set of natural numbers . The natural numbers consist of the positive whole numbers such as 1, 2, 3, 107, and 203. We will use the symbol \(\mathbb{N}\) to stand for the set of natural numbers. Another basic number system that we will be working with is the set of integers . The integers consist of zero, the positive whole numbers, and the negatives of the positive whole numbers. If \(n\) is an integer, we can write \(n = \dfrac{n}{1}\). So each integer is a rational number and hence also a real number.

We will use the letter \(\mathbb{Z}\) to stand for the set of integers. (The letter \(\mathbb{Z}\) is from the German word, \(Zahlen\), for numbers.) Three of the basic properties of the integers are that the set \(\mathbb{Z}\) is closed under addition , the set \(\mathbb{Z}\) is closed under multiplication , and the set of integers is closed under subtraction. This means that

• If \(x\) and \(y\) are integers, then \(x + y\) is an integer;
• If \(x\) and \(y\) are integers, then \(x \cdot y\) is an integer; and
• If \(x\) and \(y\) are integers, then \(x - y\) is an integer.

Notice that these so-called closure properties are defined in terms of conditional statements. This means that if we can find one instance where the hypothesis is true and the conclusion is false, then the conditional statement is false.

Example 1.6: Closure

• In order for the set of natural numbers to be closed under subtraction, the following conditional statement would have to be true: If \(x\) and \(y\) are natural numbers, then \(x - y\) is a natural number. However, since 5 and 8 are natural numbers, \(5 - 8 = -3\), which is not a natural number, this conditional statement is false. Therefore, the set of natural numbers is not closed under subtraction.
• We can use the rules for multiplying fractions and the closure rules for the integers to show that the rational numbers are closed under multiplication. If \(\dfrac{a}{b}\) and \(\dfrac{c}{d}\) are rational numbers (so \(a\), \(b\), \(c\), and \(d\) are integers and \(b\) and \(d\) are not zero), then \(\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}.\) Since the integers are closed under multiplication, we know that \(ac\) and \(bd\) are integers and since \(b \ne 0\) and \(d \ne 0\), \(bd \ne 0\). Hence, \(\dfrac{ac}{bd}\) is a rational number and this shows that the rational numbers are closed under multiplication.

Progress Check 1.7: Closure Properties

Answer each of the following questions.

• Is the set of rational numbers closed under addition? Explain.
• Is the set of integers closed under division? Explain.
• Is the set of rational numbers closed under subtraction? Explain.
• Which of the following sentences are statements? (a) \(3^2 + 4^2 = 5^2.\) (b) \(a^2 + b^2 = c^2.\) (c) There exists integers \(a\), \(b\), and \(c\) such that \(a^2 + b^2 = c^2.\) (d) If \(x^2 = 4\), then \(x = 2.\) (e) For each real number \(x\), if \(x^2 = 4\), then \(x = 2.\) (f) For each real number \(t\), \(\sin^2t + \cos^2t = 1.\) (g) \(\sin x < \sin (\frac{\pi}{4}).\) (h) If \(n\) is a prime number, then \(n^2\) has three positive factors. (i) 1 + \(\tan^2 \theta = \text{sec}^2 \theta.\) (j) Every rectangle is a parallelogram. (k) Every even natural number greater than or equal to 4 is the sum of two prime numbers.
• Identify the hypothesis and the conclusion for each of the following conditional statements. (a) If \(n\) is a prime number, then \(n^2\) has three positive factors. (b) If \(a\) is an irrational number and \(b\) is an irrational number, then \(a \cdot b\) is an irrational number. (c) If \(p\) is a prime number, then \(p = 2\) or \(p\) is an odd number. (d) If \(p\) is a prime number and \(p \ne 2\) or \(p\) is an odd number. (e) \(p \ne 2\) or \(p\) is a even number, then \(p\) is not prime.
• Determine whether each of the following conditional statements is true or false. (a) If 10 < 7, then 3 = 4. (b) If 7 < 10, then 3 = 4. (c) If 10 < 7, then 3 + 5 = 8. (d) If 7 < 10, then 3 + 5 = 8.
• Determine the conditions under which each of the following conditional sentences will be a true statement. (a) If a + 2 = 5, then 8 < 5. (b) If 5 < 8, then a + 2 = 5.
• Let \(P\) be the statement “Student X passed every assignment in Calculus I,” and let \(Q\) be the statement “Student X received a grade of C or better in Calculus I.” (a) What does it mean for \(P\) to be true? What does it mean for \(Q\) to be true? (b) Suppose that Student X passed every assignment in Calculus I and received a grade of B-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (c) Suppose that Student X passed every assignment in Calculus I and received a grade of C-, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (d) Now suppose that Student X did not pass two assignments in Calculus I and received a grade of D, and that the instructor made the statement \(P \to Q\). Would you say that the instructor lied or told the truth? (e) How are Parts ( 5b ), ( 5c ), and ( 5d ) related to the truth table for \(P \to Q\)?

Theorem If f is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and a < 0, then the function f has a maximum value when \(x = \dfrac{-b}{2a}\). Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem If \(f\) is a quadratic function of the form \(f(x) = ax^2 + bx + c\) and ac < 0, then the function \(f\) has two x-intercepts.

Using only this theorem, what can be concluded about the functions given by the following formulas? (a) \(g (x) = -8x^2 + 5x - 2\) (b) \(h (x) = -\dfrac{1}{3}x^2 + 3x\) (c) \(k (x) = 8x^2 - 5x - 7\) (d) \(j (x) = -\dfrac{71}{99}x^2 +210\) (e) \(f (x) = -4x^2 - 3x + 7\) (f) \(F (x) = -x^4 + x^3 + 9\)

Theorem A. If \(f\) is a cubic function of the form \(f (x) = x^3 - x + b\) and b > 1, then the function \(f\) has exactly one \(x\)-intercept. Following is another theorem about \(x\)-intercepts of functions: Theorem B . If \(f\) and \(g\) are functions with \(g (x) = k \cdot f (x)\), where \(k\) is a nonzero real number, then \(f\) and \(g\) have exactly the same \(x\)-intercepts.

Using only these two theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a) \(f (x) = x^3 -x + 7\) (b) \(g (x) = x^3 + x +7\) (c) \(h (x) = -x^3 + x - 5\) (d) \(k (x) = 2x^3 + 2x + 3\) (e) \(r (x) = x^4 - x + 11\) (f) \(F (x) = 2x^3 - 2x + 7\)

• (a) Is the set of natural numbers closed under division? (b) Is the set of rational numbers closed under division? (c) Is the set of nonzero rational numbers closed under division? (d) Is the set of positive rational numbers closed under division? (e) Is the set of positive real numbers closed under subtraction? (f) Is the set of negative rational numbers closed under division? (g) Is the set of negative integers closed under addition? Explorations and Activities
• Exploring Propositions . In Progress Check 1.2 , we used exploration to show that certain statements were false and to make conjectures that certain statements were true. We can also use exploration to formulate a conjecture that we believe to be true. For example, if we calculate successive powers of \(2, (2^1, 2^2, 2^3, 2^4, 2^5, ...)\) and examine the units digits of these numbers, we could make the following conjectures (among others): \(\bullet\) If \(n\) is a natural number, then the units digit of \(2^n\) must be 2, 4, 6, or 8. \(\bullet\) The units digits of the successive powers of 2 repeat according to the pattern “2, 4, 8, 6.” (a) Is it possible to formulate a conjecture about the units digits of successive powers of \(4 (4^1, 4^2, 4^3, 4^4, 4^5,...)\)? If so, formulate at least one conjecture. (b) Is it possible to formulate a conjecture about the units digit of numbers of the form \(7^n - 2^n\), where \(n\) is a natural number? If so, formulate a conjecture in the form of a conditional statement in the form “If \(n\) is a natural number, then ... .” (c) Let \(f (x) = e^(2x)\). Determine the first eight derivatives of this function. What do you observe? Formulate a conjecture that appears to be true. The conjecture should be written as a conditional statement in the form, “If n is a natural number, then ... .”

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If-then statement

• Logical correct I
• Logical correct II

When we previously discussed inductive reasoning we based our reasoning on examples and on data from earlier events. If we instead use facts, rules and definitions then it's called deductive reasoning.

We will explain this by using an example.

If you get good grades then you will get into a good college.

The part after the "if": you get good grades - is called a hypotheses and the part after the "then" - you will get into a good college - is called a conclusion.

Hypotheses followed by a conclusion is called an If-then statement or a conditional statement.

This is noted as

$$p \to q$$

This is read - if p then q.

A conditional statement is false if hypothesis is true and the conclusion is false. The example above would be false if it said "if you get good grades then you will not get into a good college".

If we re-arrange a conditional statement or change parts of it then we have what is called a related conditional.

Our conditional statement is: if a population consists of 50% men then 50% of the population must be women.

If we exchange the position of the hypothesis and the conclusion we get a converse statemen t: if a population consists of 50% women then 50% of the population must be men.

$$q\rightarrow p$$

If both statements are true or if both statements are false then the converse is true. A conditional and its converse do not mean the same thing

If we negate both the hypothesis and the conclusion we get a inverse statemen t: if a population do not consist of 50% men then the population do not consist of 50% women.

$$\sim p\rightarrow \: \sim q$$

The inverse is not true juest because the conditional is true. The inverse always has the same truth value as the converse.

We could also negate a converse statement, this is called a contrapositive statemen t:  if a population do not consist of 50% women then the population do not consist of 50% men.

$$\sim q\rightarrow \: \sim p$$

The contrapositive does always have the same truth value as the conditional. If the conditional is true then the contrapositive is true.

A pattern of reaoning is a true assumption if it always lead to a true conclusion. The most common patterns of reasoning are detachment and syllogism.

If we turn of the water in the shower, then the water will stop pouring.

If we call the first part p and the second part q then we know that p results in q. This means that if p is true then q will also be true. This is called the law of detachment and is noted:

$$\left [ (p \to q)\wedge p \right ] \to q$$

The law of syllogism tells us that if p → q and q → r then p → r is also true.

This is noted:

$$\left [ (p \to q)\wedge (q \to r ) \right ] \to (p \to r)$$

If the following statements are true:

If we turn of the water (p), then the water will stop pouring (q). If the water stops pouring (q) then we don't get wet any more (r).

Then the law of syllogism tells us that if we turn of the water (p) then we don't get wet (r) must be true.

Video lesson

Write a converse, inverse and contrapositive to the conditional

"If you eat a whole pint of ice cream, then you won't be hungry"

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Logic and Mathematical Statements

Worked examples, if...then... statements, mini-lecture., example. consider the statement "$x > 0 \rightarrow x+1>0$". is this statement true or false, example. consider the statement "if $x$ is a positive integer or a solution to $x+3>4$, then $x>0$ and $x> \frac{1}{2}$." is this statement true, example. consider the statement "$0>1 \rightarrow \sin x = 2$". is this statement true or false.

Understanding the Role of Hypotheses and Conclusions in Mathematical Reasoning

Hypothesis and conclusion.

In the context of mathematics and logic, a hypothesis is a statement or proposition that is assumed to be true for the purpose of a logical argument or investigation. It is usually denoted by “H” or “P” and is the starting point for many mathematical proofs.

For example, let’s consider the hypothesis: “If it is raining outside, then the ground is wet.” This statement assumes that whenever it rains, the ground will be wet.

The conclusion, on the other hand, is the statement or proposition that is inferred or reached by logical reasoning, based on the hypothesis or given information. It is typically denoted by “C” or “Q”.

Using the same example, the conclusion derived from the hypothesis could be: “It is currently raining outside, so the ground is wet.” This conclusion is based on the assumption that the given condition of rain implies a wet ground.

In mathematics, hypotheses and conclusions are commonly used in proofs and logical arguments. By stating a hypothesis and then deducing a conclusion from it, mathematicians can demonstrate the validity of certain mathematical concepts, theorems, or formulas.

It’s important to note that in mathematics, a hypothesis is not the same as a guess or a prediction. It is a statement that is assumed to be true and serves as the basis for logical reasoning, while the conclusion is the logical consequence or outcome that is drawn from the hypothesis.

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Conditional Statement If Then's Defined in Geometry - 15+ Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson , you’re going to learn all about conditional statements!

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’re going to walk through several examples to ensure you know what you’re doing.

In addition, this lesson will prepare you for deductive reasoning and two column proofs later on.

Here we go!

What are Conditional Statements?

To better understand deductive reasoning, we must first learn about conditional statements.

A conditional statement has two parts: hypothesis ( if ) and conclusion ( then ).

In fact, conditional statements are nothing more than “If-Then” statements!

Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.

But to verify statements are correct, we take a deeper look at our if-then statements. This is why we form the converse , inverse , and contrapositive of our conditional statements.

What is the Converse of a Statement?

Well, the converse is when we switch or interchange our hypothesis and conclusion.

Conditional Statement : “If today is Wednesday, then yesterday was Tuesday.”

Hypothesis : “If today is Wednesday” so our conclusion must follow “Then yesterday was Tuesday.”

So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states.

Converse : “If yesterday was Tuesday, then today is Wednesday.”

What is the Inverse of a Statement?

Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional statement.

So using our current conditional statement, “If today is Wednesday, then yesterday was Tuesday”.

Inverse : “If today is not Wednesday, then yesterday was not Tuesday.”

What is a Contrapositive?

And the contrapositive is formed by interchanging the hypothesis and conclusion and then negating both.

Contrapositive : “If yesterday was not Tuesday, then today is not Wednesday”

What is a Biconditional Statement?

A statement written in “if and only if” form combines a reversible statement and its true converse. In other words the conditional statement and converse are both true.

Continuing with our initial condition, “If today is Wednesday, then yesterday was Tuesday.”

Biconditional : “Today is Wednesday if and only if yesterday was Tuesday.”

Examples of Conditional Statements

In the video below we will look at several harder examples of how to form a proper statement, converse, inverse, and contrapositive. And here’s a big hint…

Whenever you see “con” that means you switch! It’s like being a con-artist!

Moreover, we will detail the process for coming up with reasons for our conclusions using known postulates. We will review the ten postulates that we have learned so far, and add a few more problems dealing with perpendicular lines, planes, and perpendicular bisectors.

After this lesson, we will be ready to tackle deductive reasoning head-on, and feel confident as we march onward toward learning two-column proofs!

Conditional Statements – Lesson & Examples (Video)

• Introduction to conditional statements
• 00:00:25 – What are conditional statements, converses, and biconditional statements? (Examples #1-2)
• 00:05:21 – Understanding venn diagrams (Examples #3-4)
• 00:11:07 – Supply the missing venn diagram and conditional statement for each question (Examples #5-8)
• Exclusive Content for Member’s Only
• 00:17:48 – Write the statement and converse then determine if they are reversible (Examples #9-12)
• 00:29:17 – Understanding the inverse, contrapositive, and symbol notation
• 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14)
• 00:45:40 – Using geometry postulates to verify statements (Example #15)
• 00:53:23 – What are perpendicular lines, perpendicular planes and the perpendicular bisector?
• 00:56:26 – Using the figure, determine if the statement is true or false (Example #16)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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Conditional Statement – Definition, Truth Table, Examples, FAQs

What is a conditional statement, how to write a conditional statement, what is a biconditional statement, solved examples on conditional statements, practice problems on conditional statements, frequently asked questions about conditional statements.

A conditional statement is a statement that is written in the “If p, then q” format. Here, the statement p is called the hypothesis and q is called the conclusion. It is a fundamental concept in logic and mathematics. 

Conditional statement symbol :  p → q

A conditional statement consists of two parts.

• The “if” clause, which presents a condition or hypothesis.
• The “then” clause, which indicates the consequence or result that follows if the condition is true. 

Example : If you brush your teeth, then you won’t get cavities.

Hypothesis (Condition): If you brush your teeth

Conclusion (Consequence): then you won’t get cavities 

Conditional Statement: Definition

A conditional statement is characterized by the presence of “if” as an antecedent and “then” as a consequent. A conditional statement, also known as an “if-then” statement consists of two parts:

• The “if” clause (hypothesis): This part presents a condition, situation, or assertion. It is the initial condition that is being considered.
• The “then” clause (conclusion): This part indicates the consequence, result, or action that will occur if the condition presented in the “if” clause is true or satisfied. 

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Representation of Conditional Statement

The conditional statement of the form ‘If p, then q” is represented as p → q. 

It is pronounced as “p implies q.”

Different ways to express a conditional statement are:

• p implies q
• p is sufficient for q
• q is necessary for p

Parts of a Conditional Statement

There are two parts of conditional statements, hypothesis and conclusion. The hypothesis or condition will begin with the “if” part, and the conclusion or action will begin with the “then” part. A conditional statement is also called “implication.”

Conditional Statements Examples:

Example 1: If it is Sunday, then you can go to play. 

Hypothesis: If it is Sunday

Conclusion: then you can go to play. 

Example 2: If you eat all vegetables, then you can have the dessert.

Condition: If you eat all vegetables

Conclusion: then you can have the dessert 

To form a conditional statement, follow these concise steps:

Step 1 : Identify the condition (antecedent or “if” part) and the consequence (consequent or “then” part) of the statement.

Step 2 : Use the “if… then…” structure to connect the condition and consequence.

Step 3 : Ensure the statement expresses a logical relationship where the condition leads to the consequence.

Example 1 : “If you study (condition), then you will pass the exam (consequence).” 

This conditional statement asserts that studying leads to passing the exam. If you study (condition is true), then you will pass the exam (consequence is also true).

Example 2 : If you arrange the numbers from smallest to largest, then you will have an ascending order.

Hypothesis: If you arrange the numbers from smallest to largest

Conclusion: then you will have an ascending order

Truth Table for Conditional Statement

The truth table for a conditional statement is a table used in logic to explore the relationship between the truth values of two statements. It lists all possible combinations of truth values for “p” and “q” and determines whether the conditional statement is true or false for each combination. 

The truth value of p → q is false only when p is true and q is False. 

If the condition is false, the consequence doesn’t affect the truth of the conditional; it’s always true.

In all the other cases, it is true.

The truth table is helpful in the analysis of possible combinations of truth values for hypothesis or condition and conclusion or action. It is useful to understand the presence of truth or false statements. 

Converse, Inverse, and Contrapositive

The converse, inverse, and contrapositive are three related conditional statements that are derived from an original conditional statement “p → q.” 

Consider a conditional statement: If I run, then I feel great.

• Converse: 

The converse of “p → q” is “q → p.” It reverses the order of the original statement. While the original statement says “if p, then q,” the converse says “if q, then p.” 

Converse: If I feel great, then I run.

• Inverse: 

The inverse of “p → q” is “~p → ~q,” where “” denotes negation (opposite). It negates both the antecedent (p) and the consequent (q). So, if the original statement says “if p, then q,” the inverse says “if not p, then not q.”

Inverse : If I don’t run, then I don’t feel great.

• Contrapositive: 

The contrapositive of “p → q” is “~q → ~p.” It reverses the order and also negates both the statements. So, if the original statement says “if p, then q,” the contrapositive says “if not q, then not p.”

Contrapositive: If I don’t feel great, then I don’t run.

A biconditional statement is a type of compound statement in logic that expresses a bidirectional or two-way relationship between two statements. It asserts that “p” is true if and only if “q” is true, and vice versa. In symbolic notation, a biconditional statement is represented as “p ⟺ q.”

In simpler terms, a biconditional statement means that the truth of “p” and “q” are interdependent. 

If “p” is true, then “q” must also be true, and if “q” is true, then “p” must be true. Conversely, if “p” is false, then “q” must be false, and if “q” is false, then “p” must be false. 

Biconditional statements are often used to express equality, equivalence, or conditions where two statements are mutually dependent for their truth values. 

Examples : 

• I will stop my bike if and only if the traffic light is red.  
• I will stay if and only if you play my favorite song.

Facts about Conditional Statements

• The negation of a conditional statement “p → q” is expressed as “p and not q.” It is denoted as “𝑝 ∧ ∼𝑞.” 
• The conditional statement is not logically equivalent to its converse and inverse.
• The conditional statement is logically equivalent to its contrapositive. 
• Thus, we can write p → q ∼q → ∼p

In this article, we learned about the fundamentals of conditional statements in mathematical logic, including their structure, parts, truth tables, conditional logic examples, and various related concepts. Understanding conditional statements is key to logical reasoning and problem-solving. Now, let’s solve a few examples and practice MCQs for better comprehension.

Example 1: Identify the hypothesis and conclusion. 

If you sing, then I will dance.

Solution : 

Given statement: If you sing, then I will dance.

Here, the antecedent or the hypothesis is “if you sing.”

The conclusion is “then I will dance.”

Example 2: State the converse of the statement: “If the switch is off, then the machine won’t work.” 

Here, p: The switch is off

q: The machine won’t work.

The conditional statement can be denoted as p → q.

Converse of p → q is written by reversing the order of p and q in the original statement.

Converse of  p → q is q → p.

Converse of  p → q: q → p: If the machine won’t work, then the switch is off.

Example 3: What is the truth value of the given conditional statement? 

If 2+2=5 , then pigs can fly.

Solution:  

q: Pigs can fly.

The statement p is false. Now regardless of the truth value of statement q, the overall statement will be true. 

F → F = T

Hence, the truth value of the statement is true. 

Conditional Statement - Definition, Truth Table, Examples, FAQs

Attend this quiz & Test your knowledge.

What is the antecedent in the given conditional statement? If it’s sunny, then I’ll go to the beach.

A conditional statement can be expressed as, what is the converse of “a → b”, when the antecedent is true and the consequent is false, the conditional statement is.

What is the meaning of conditional statements?

Conditional statements, also known as “if-then” statements, express a cause-and-effect or logical relationship between two propositions.

When does the truth value of a conditional statement is F?

A conditional statement is considered false when the antecedent is true and the consequent is false.

What is the contrapositive of a conditional statement?

The contrapositive reverses the order of the statements and also negates both the statements. It is equivalent in truth value to the original statement.

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Understanding Hypotheses

'What happens if ... ?' to ' This will happen if'

The experimentation of children continually moves on to the exploration of new ideas and the refinement of their world view of previously understood situations. This description of the playtime patterns of young children very nicely models the concept of 'making and testing hypotheses'. It follows this pattern:

• Make some observations. Collect some data based on the observations.
• Draw a conclusion (called a 'hypothesis') which will explain the pattern of the observations.
• Test out your hypothesis by making some more targeted observations.

So, we have

• A hypothesis is a statement or idea which gives an explanation to a series of observations.

Sometimes, following observation, a hypothesis will clearly need to be refined or rejected. This happens if a single contradictory observation occurs. For example, suppose that a child is trying to understand the concept of a dog. He reads about several dogs in children's books and sees that they are always friendly and fun. He makes the natural hypothesis in his mind that dogs are friendly and fun . He then meets his first real dog: his neighbour's puppy who is great fun to play with. This reinforces his hypothesis. His cousin's dog is also very friendly and great fun. He meets some of his friends' dogs on various walks to playgroup. They are also friendly and fun. He is now confident that his hypothesis is sound. Suddenly, one day, he sees a dog, tries to stroke it and is bitten. This experience contradicts his hypothesis. He will need to amend the hypothesis. We see that

• Gathering more evidence/data can strengthen a hypothesis if it is in agreement with the hypothesis.
• If the data contradicts the hypothesis then the hypothesis must be rejected or amended to take into account the contradictory situation.

• A contradictory observation can cause us to know for certain that a hypothesis is incorrect.
• Accumulation of supporting experimental evidence will strengthen a hypothesis but will never let us know for certain that the hypothesis is true.

In short, it is possible to show that a hypothesis is false, but impossible to prove that it is true!

Whilst we can never prove a scientific hypothesis to be true, there will be a certain stage at which we decide that there is sufficient supporting experimental data for us to accept the hypothesis. The point at which we make the choice to accept a hypothesis depends on many factors. In practice, the key issues are

• What are the implications of mistakenly accepting a hypothesis which is false?
• What are the cost / time implications of gathering more data?
• What are the implications of not accepting in a timely fashion a true hypothesis?

For example, suppose that a drug company is testing a new cancer drug. They hypothesise that the drug is safe with no side effects. If they are mistaken in this belief and release the drug then the results could have a disastrous effect on public health. However, running extended clinical trials might be very costly and time consuming. Furthermore, a delay in accepting the hypothesis and releasing the drug might also have a negative effect on the health of many people.

In short, whilst we can never achieve absolute certainty with the testing of hypotheses, in order to make progress in science or industry decisions need to be made. There is a fine balance to be made between action and inaction.

Hypotheses and mathematics So where does mathematics enter into this picture? In many ways, both obvious and subtle:

• A good hypothesis needs to be clear, precisely stated and testable in some way. Creation of these clear hypotheses requires clear general mathematical thinking.
• The data from experiments must be carefully analysed in relation to the original hypothesis. This requires the data to be structured, operated upon, prepared and displayed in appropriate ways. The levels of this process can range from simple to exceedingly complex.

Very often, the situation under analysis will appear to be complicated and unclear. Part of the mathematics of the task will be to impose a clear structure on the problem. The clarity of thought required will actively be developed through more abstract mathematical study. Those without sufficient general mathematical skill will be unable to perform an appropriate logical analysis.

Using deductive reasoning in hypothesis testing

There is often confusion between the ideas surrounding proof, which is mathematics, and making and testing an experimental hypothesis, which is science. The difference is rather simple:

• Mathematics is based on deductive reasoning : a proof is a logical deduction from a set of clear inputs.
• Science is based on inductive reasoning : hypotheses are strengthened or rejected based on an accumulation of experimental evidence.

Of course, to be good at science, you need to be good at deductive reasoning, although experts at deductive reasoning need not be mathematicians. Detectives, such as Sherlock Holmes and Hercule Poirot, are such experts: they collect evidence from a crime scene and then draw logical conclusions from the evidence to support the hypothesis that, for example, Person M. committed the crime. They use this evidence to create sufficiently compelling deductions to support their hypotheses beyond reasonable doubt . The key word here is 'reasonable'. There is always the possibility of creating an exceedingly outlandish scenario to explain away any hypothesis of a detective or prosecution lawyer, but judges and juries in courts eventually make the decision that the probability of such eventualities are 'small' and the chance of the hypothesis being correct 'high'.

• If a set of data is normally distributed with mean 0 and standard deviation 0.5 then there is a 97.7% certainty that a measurement will not exceed 1.0.
• If the mean of a sample of data is 12, how confident can we be that the true mean of the population lies between 11 and 13?

It is at this point that making and testing hypotheses becomes a true branch of mathematics. This mathematics is difficult, but fascinating and highly relevant in the information-rich world of today.

To read more about the technical side of hypothesis testing, take a look at What is a Hypothesis Test?

You might also enjoy reading the articles on statistics on the Understanding Uncertainty website

This resource is part of the collection Statistics - Maths of Real Life

Professor: Erika L.C. King Email: [email protected] Office: Lansing 304 Phone: (315)781-3355

The majority of statements in mathematics can be written in the form: "If A, then B." For example: "If a function is differentiable, then it is continuous". In this example, the "A" part is "a function is differentiable" and the "B" part is "a function is continuous." The "A" part of the statement is called the "hypothesis", and the "B" part of the statement is called the "conclusion". Thus the hypothesis is what we must assume in order to be positive that the conclusion will hold.

Whenever you are asked to state a theorem, be sure to include the hypothesis. In order to know when you may apply the theorem, you need to know what constraints you have. So in the example above, if we know that a function is differentiable, we may assume that it is continuous. However, if we do not know that a function is differentiable, continuity may not hold. Some theorems have MANY hypotheses, some of which are written in sentences before the ultimate "if, then" statement. For example, there might be a sentence that says: "Assume n is even." which is then followed by an if,then statement. Include all hypotheses and assumptions when asked to state theorems and definitions!

Still have questions? Please ask!

Look at the example statement below:

If you take a class in TV broadcasting, then you will film a sporting event.

In this statement, the hypothesis is “taking a class in TV broadcasting”, and the conclusion is “you will film a sporting event”.

Take a look at another example:

If a number is even, then it is divisible by two.

In this example, the “if” part is the hypothesis : if a number is even

And the “then” part is the conclusion : then it is divisible by two.

Here are some more examples, and you can identify the hypothesis and conclusion by yourself:

If it is the 4 th of July, then it is a holiday (in US).

If an animal lives in the water, then it is a fish.

You are a US citizen if you are a member of the Congress.

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Hypothesis and Conclusion

If-then statement, a implies b, conclusion|definition & meaning.

 The term conclusion in maths is used to define us about the problem that we solve and when we produce the final result at the end then that stage of processes is called as conclusion.

Figure 1 – Give the Right Conclusion to the problem 

When you solve a maths question, you have to end the problem by calculating the last answer and pulling a conclus ion by writing the answer.  A conclusion is the last step of the maths problem. The conclusion is the final answer produced in the end . The answer is completed by writing the arguments and statements by telling the answer to the question. The ending statement of a problem is called a conclusion.

Drawing conclusions refers to the act of thinking of interpreting a series of premises or some ideas and, from them, suggesting something that leads to a meaningful finding. It is normally regarded as a conscious way of learning .

As a rule, a mathematical statement comprises two sections : the first section is assumptions or hypotheses , and the other section is the conclusion . Most mathematical statements have the form “If A, then B.” Often, this statement is written as “A implies B” or “A $\Rightarrow $ B.”  The assumptions we make are what makes “A,” and the circumstances that make “B” are called the conclusion .

To prove that a given statement “If A, then B” is said to be true, we will require some assumptions for “A,” and after doing some work on it, we need to conclude that “B” must also hold when “A” holds.

If we are asked to apply the statement “If A, then B,” firstly, we should be sure that the conditions of the statement “A” are met and true before we start to talk about the conclusion “B.”

Suppose you want to apply the statement “x is even $\Rightarrow$ x2 is an integer.” First, you must verify  that x is even  before  you  conclude that x2 is an integer.

In maths, you will, at many times, confront statements in the form “X $\Leftrightarrow$ Y” or “X if and only if Y.”  These statements are actually two “if, then” statements. The following statement, “X if and only if Y,” is logically equivalent to the statements “If X, then Y” and “If Y, then X.” One more method for thinking about this kind of explanation is an equality between the statements X and Y: so, whenever X holds, Y holds, and whenever Y hold, X holds.

Assume the example: “ x is even $\Leftrightarrow$ x 2  is an integer “. Statement A says, “ x is even,” whereas statement B says, “ x 2  is an integer.” If we get a quick revision about what it suggests to be even (simply that x is a multiple of 2), we can see with ease that the following two statements are identical : If x = 2 k is proved to be even, then it implies x 2 = 2 k 2 = k is an integer, and we know that x 2 = k is an integer, then x = 2 k so n is proved to be even.

In day-to-day use, a statement which is in the form “ If A, then B ,” in some cases, means “ A if and only if B. ” For example, when people agree on a deal, they say, “If you agree to sell me your car for 500k, then I’ll buy from you this week” they straightaway mean, “I’ll buy your car if and only if you agree to sell me in 500k.” In other words, if you don’t agree on 500k, they will not be buying your car from you .

In geometry, the validation or proof is stated in the if-then format. The “if” is a condition or hypothesis , and if that condition is met, only then the second part of the statement is true , which is called the conclusion . The working is like any other if-then statement. For illustration, the statement “If a toy shop has toys for two age groups and 45 percent of toys in the shop are for 14 or above years old, then 55 percent of the toys in the shop are for 13 and fewer years old.” The above statement concludes that “55 percent of the toys in the shop are for 13 and fewer years old.”

In maths, the statement “A if and only if B” is very different from “A implies B.” Assume the example: “ x is an integer” is the A statement, and “ x 3 is a rational number” is the B statement  The statement “A implies B” here means “If x is an integer, then x 3 is a rational number.” The statement is proven to be true. On the other hand, the statement, “A, if and only if B,” means “ x is an integer if and only if x 3 is a rational number,” which is not true in this case.

Examples of Drawing Conclusions

Consider the equation below. Comment if this equation is true or false.

Figure 3 – Example Problem

To calculate its true answer, first, consider the hypothesis $x>0$. Whatever we are going to conclude, it will be a consequence of the truth that $x$ is positive.

Next, consider the conclusion $x+1>0$. This equation is right, since $x+1>x>0$.

This implies that the provided inequality is true.

Simplify the below problem by providing a conclusion by calculating the answer of A.

\[ A= \dfrac{35}{3} \]

The expression given in the question is: $A= \dfrac{35}{3}$

Calculating the answer of A to make a conclusion, The arithmetic operation division is found in the question that is to be figured out in the provided problem. After figuring out the answer to expression A, The conclusion will be given.

\[ A= 11. 667 \]

Therefore, we conclude the question by calculating the answer of $A=11.666$

Consider the equation $0>1 \Rightarrow sinx=2$. Is this equation true or false?

To calculate the correct answer, first consider the hypothesis $0>1$. This equation is clearly false.

calculate the below problem by providing a conclusion by estimating the value of X.

\[ 3+8 \times 2\]

The expression given in the problem is $3+8 \times 2 $.

Multiplication and Plus operation is to be carried out to calculate the answer to the given problem. After figuring out the answer to X  the conclusion will be given.

Thus, we conclude the example by calculating the value of $X = 19$.

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

Statistics Made Easy

How to Write Hypothesis Test Conclusions (With Examples)

A   hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

• Null Hypothesis (H 0 ): The sample data occurs purely from chance.
• Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we reject the null hypothesis .

Otherwise, if the p-value is not less than some significance level then we fail to reject the null hypothesis .

When writing the conclusion of a hypothesis test, we typically include:

• Whether we reject or fail to reject the null hypothesis.
• The significance level.
• A short explanation in the context of the hypothesis test.

For example, we would write:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that…

Or, we would write:

We fail to reject the null hypothesis at the 5% significance level.   There is not sufficient evidence to support the claim that…

The following examples show how to write a hypothesis test conclusion in both scenarios.

Example 1: Reject the Null Hypothesis Conclusion

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

• H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
• H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

Example 2: Fail to Reject the Null Hypothesis Conclusion

Suppose the manager of a manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, he measures the mean number of defective widgets produced before and after using the new method for one month.

He performs a hypothesis test at a 10% significance level using the following hypotheses:

• H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
• H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We fail to reject the null hypothesis at the 10% significance level.   There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing 4 Examples of Hypothesis Testing in Real Life How to Write a Null Hypothesis

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

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Write a conditional with the given hypothesis and conclusion. Hypothesis: A number is a prime. Conclusion: The number has exactly two divisors. B I U

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