hypothesis of the following conditional statement

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. 

Related Worksheets

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.

RELATED POSTS

• Ordering Decimals: Definition, Types, Examples
• Decimal to Octal: Steps, Methods, Conversion Table
• Lattice Multiplication – Definition, Method, Examples, Facts, FAQs
• X Intercept – Definition, Formula, Graph, Examples
• Lateral Face – Definition With Examples

Math & ELA | PreK To Grade 5

Kids see fun., you see real learning outcomes..

Make study-time fun with 14,000+ games & activities, 450+ lesson plans, and more—free forever.

Parents, Try for Free Teachers, Use for Free

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.11: If Then Statements

• Last updated
• Save as PDF
• Page ID 2144

Hypothesis followed by a conclusion in a conditional statement.

Conditional Statements

A conditional statement (also called an if-then statement ) is a statement with a hypothesis followed by a conclusion . The hypothesis is the first, or “if,” part of a conditional statement. The conclusion is the second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis.

If-then statements might not always be written in the “if-then” form. Here are some examples of conditional statements:

• Statement 1: If you work overtime, then you’ll be paid time-and-a-half.
• Statement 2: I’ll wash the car if the weather is nice.
• Statement 3: If 2 divides evenly into \(x\), then \(x\) is an even number.
• Statement 4: I’ll be a millionaire when I win the lottery.
• Statement 5: All equiangular triangles are equilateral.

Statements 1 and 3 are written in the “if-then” form. The hypothesis of Statement 1 is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.” Statement 2 has the hypothesis after the conclusion. If the word “if” is in the middle of the statement, then the hypothesis is after it. The statement can be rewritten: If the weather is nice, then I will wash the car. Statement 4 uses the word “when” instead of “if” and is like Statement 2. It can be written: If I win the lottery, then I will be a millionaire. Statement 5 “if” and “then” are not there. It can be rewritten: If a triangle is equiangular, then it is equilateral.

What if you were given a statement like "All squares are rectangles"? How could you determine the hypothesis and conclusion of this statement?

Example \(\PageIndex{1}\)

Determine the hypothesis and conclusion: I'll bring an umbrella if it rains.

Hypothesis: "It rains." Conclusion: "I'll bring an umbrella."

Example \(\PageIndex{2}\)

Determine the hypothesis and conclusion: All right angles are \(90^{\circ}\).

Hypothesis: "An angle is right." Conclusion: "It is \(90^{\circ}\)."

Example \(\PageIndex{3}\)

Use the statement: I will graduate when I pass Calculus.

Rewrite in if-then form and determine the hypothesis and conclusion.

This statement can be rewritten as If I pass Calculus, then I will graduate. The hypothesis is “I pass Calculus,” and the conclusion is “I will graduate.”

Example \(\PageIndex{4}\)

Use the statement: All prime numbers are odd.

Rewrite in if-then form, determine the hypothesis and conclusion, and determine whether this is a true statement.

This statement can be rewritten as If a number is prime, then it is odd. The hypothesis is "a number is prime" and the conclusion is "it is odd". This is not a true statement (remember that not all conditional statements will be true!) since 2 is a prime number but it is not odd.

Example \(\PageIndex{5}\)

Determine the hypothesis and conclusion: Sarah will go to the store if Riley does the laundry.

The statement can be rewritten as "If Riley does the laundry then Sarah will go to the store." The hypothesis is "Riley does the laundry" and the conclusion is "Sarah will go to the store."

Determine the hypothesis and the conclusion for each statement.

• If 5 divides evenly into \(x\), then \(x\) ends in 0 or 5.
• If a triangle has three congruent sides, it is an equilateral triangle.
• Three points are coplanar if they all lie in the same plane.
• If \(x=3\), then \(x^2=9\).
• If you take yoga, then you are relaxed.
• All baseball players wear hats.
• I'll learn how to drive when I am 16 years old.
• If you do your homework, then you can watch TV.
• Alternate interior angles are congruent if lines are parallel.
• All kids like ice cream.

Additional Resources

Video: If-Then Statements Principles - Basic

Activities: If-Then Statements Discussion Questions

Study Aids: Conditional Statements Study Guide

Practice: If Then Statements

Real World: If Then Statements

Talk to our experts

1800-120-456-456

• Conditional Statement

What Is A Conditional Statement?

In mathematics, we define statement as a declarative statement which may either be true or may be false. Often sentences that are mathematical in nature may not be a statement because we might not know what the variable represents. For example, 2x + 2 = 5. Now here we do not know what x represents thus if we substitute the value of x (let us consider that x = 3) i.e., 2 × 3 = 6. Therefore, it is a false statement. So, what is a conditional statement? In simple words, when through a statement we put a condition on something in return of something, we call it a conditional statement. For example, Mohan tells his friend that “if you do my homework, then I will pay you 50 dollars”. So what is happening here? Mohan is paying his friend 50 dollars but places a condition that if only he’s work will be completed by his friend. A conditional statement is made up of two parts. First, there is a hypothesis that is placed after “if” and before the comma and second is a conclusion that is placed after “then”. Here, the hypothesis will be “you do my homework” and the conclusion will be “I will pay you 50 dollars”. Now, this statement can either be true or may be false. We don’t know. 

A hypothesis is a part that is used after the 'if' and before the comma. This composes the first part of a conditional statement. For example, the statement, 'I help you get an A+ in math,' is a hypothesis because this phrase is coming in between the 'if' and the comma. So, now I hope you can spot the hypothesis in other examples of a conditional statement. Of course, you can. Here is a statement: 'If Miley gets a car, then Allie's dog will be trained,' the hypothesis here is, 'Miley gets a car.' For the statement, 'If Tom eats chocolate ice cream, then Luke eats double chocolate ice cream,' the hypothesis here is, 'Tom eats chocolate ice cream. Now it is time for you to try and locate the hypothesis for the statement, 'If the square is a rectangle, then the rectangle is a quadrilateral'?

A conclusion is a part that is used after “then”. This composes the second part of a conditional statement. For example, for the statement, “I help you get an A+ in math”, the conclusion will be “you will give me 50 dollars”. The next statement was “If Miley gets a car, then Allie's dog will be trained”, the conclusion here is Allie's dog will be trained. It is the same with the next statement and for every other conditional statement.   

How Do We Know If A Statement Is True or False? 

In mathematics, the best way we can know if a statement is true or false is by writing a mathematical proof. Before writing a proof, the mathematician must find if the statement is true or false that can be done with the help of exploration and then by finding the counterexample. Once the proof is discovered, the mathematician must communicate this discovery to those who speak the language of maths. 

Converse, Inverse, contrapositive, And Bi-conditional Statement

We usually use the term “converse” as a verb for talking and chatting and as a noun we use it to represent a brand of footwear. But in mathematics, we use it differently. Converse and inverse are the two terms that are a connected concept in the making of a conditional statement.

If we want to create the converse of a conditional statement, we just have to switch the hypothesis and the conclusion. To create the inverse of a conditional statement, we have to turn both the hypothesis and the conclusion to the negative. A contrapositive statement can be made if we first interchange the hypothesis and conclusion then make them both negative. In a bi-conditional statement, we use “if and only if” which means that the hypothesis is true only if the condition is true. For example, 

If you eat junk food, then you will gain weight is a conditional statement.

If you gained weight, then you ate junk food is a converse of a conditional statement.

If you do not eat junk food, then you will not gain weight is an inverse of a conditional statement.

If yesterday was not Monday, then today is not Tuesday is a contrapositive statement. 

Today is Tuesday if and only if yesterday was Monday is a bi-conventional statement.   

Image will be uploaded soon

A Conditional Statement Truth Table

In the table above, p→q will be false only if the hypothesis(p) will be true and the conclusion(q) will be false, or else p→q will be true. 

Conditional Statement Examples

Below, you can see some of the conditional statement examples.

Example 1) Given, P = I do my work; Q = I get the allowance

What does p→q represent?

Solution 1) In the sentence above, the hypothesis is “I do my work” and the conclusion is “ I get the allowance”. Therefore, the condition p→q represents the conditional statement, “If I do my work, then I get the allowance”. 

Example 2) Given, a = The sun is a ball of gas; b = 5 is a prime number. Write a→b in a sentence. 

Solution 2) The conditional statement a→b here is “if the sun is a ball of gas, then 5 is a prime number”.

FAQs on Conditional Statement

1. How many types of conditional statements are there?

There are basically 5 types of conditional statements.

If statement, if-else statement, nested if-else statement, if-else-if ladder, and switch statement are the basic types of conditional statements. If a function displays a statement or performs a function on the condition if the statement is true. If-else statement executes a block of code if the condition is true but if the condition is false, a new block of code is placed. The switch statement is a selection control mechanism that allows the value of a variable to change the control flow of a program. 

2. How are a conditional statement and a loop different from each other?

A conditional statement is sometimes used by a loop but a loop is of no use to a conditional statement. A conditional statement is basically a “yes” or a “no” i.e., if yes, then take the first path else take the second one. A loop is more like a cyclic chain starting from the start point i.e., if yes, then take path a, if no, take path b and it comes back to the start point. 

Conditional statement executes a statement based on a condition without causing any repetition. 

A loop executes a statement repeatedly. There are two loop variables i.e., for loop and while loop.

Become a math whiz with AI Tutoring, Practice Questions & more.

Conditional Statements

Two common types of statements found in the study of logic are conditional and biconditional statements. They are formed by combining two statements which we then we call compound statements. What if we were to say, "If it snows, then we don't go outside." This is two statements combined. They are often called if-then statements. As in, "IF it snows, THEN we don't go outside." They are a fundamental building block of computer programming.

Writing conditional statements

A statement written in if-then format is a conditional statement.

It looks like

This represents the conditional statement:

"If p then q ."

A conditional statement is also called an implication.

If a closed shape has three sides, then it is a triangle.

The part of the statement that follows the "if" is called the hypothesis. The part of the statement that follows the "then" is the conclusion.

So in the above statement,

If a closed shape has three sides, (this is the hypothesis)

Then it is a triangle. (this is the conclusion)

Identify the hypothesis and conclusion of the following conditional statement.

A polygon is a hexagon if it has six sides.

Hypothesis: The polygon has six sides.

Conclusion: It is a hexagon.

The hypothesis does not always come first in a conditional statement. You must read it carefully to determine which part of the statement is the hypothesis and which part is the conclusion.

Truth table for conditional statement

The truth table for any two given inputs, say A and B , is given by:

• If A and B are both true, then A → B is true.
• If A is true and B is false, then A → B is false.
• If A is false and B is true, then A → B is true.
• If A and B are both false, then A → B is true.

Take our conditional statement that if it snows, we do not go outside.

If it is snowing ( A is true) and we do go outside ( B is false), then the statement A → B is false.

If it is not snowing ( A is false), it doesn't matter if we go outside or not ( B is true or false), because A → B is impossible to determine if A is false, so the statement A → B can still be true.

Biconditional statements

A biconditional statement is a combination of a statement and its opposite written in the format of "if and only if."

For example, "Two line segments are congruent if and only if they are the same length."

This is a combination of two conditional statements.

"Two line segments are congruent if they are the same length."

"Two line segments are the same length if they are congruent."

A biconditional statement is true if and only if both the conditional statements are true.

Biconditional statements are represented by the symbol:

p ↔ q

p ↔ q = p → q ∧ q → p

Writing biconditional statements

Write the two conditional statements that make up this biconditional statement:

I am punctual if and only if I am on time to school every day.

The two conditional statements that have to be true to make this statement true are:

• I am punctual if I am on time to school every day.
• I am on time to school every day if I am punctual.

A rectangle is a square if and only if the adjacent sides are congruent.

• If the adjacent sides of a rectangle are congruent then it is a square.
• If a rectangle is a square then the adjacent sides are congruent.

Topics related to the Conditional Statements

Conjunction

Counter Example

Biconditional Statement

Flashcards covering the Conditional Statements

Symbolic Logic Flashcards

Introduction to Proofs Flashcards

Practice tests covering the Conditional Statements

Introduction to Proofs Practice Tests

Get help learning about conditional statements

Understanding conditional statements can be tricky, especially when it gets deeper into programming language. If your student needs a boost in their comprehension of conditional statements, have them meet with an expert tutor who can give them 1-on-1 support in a setting free from distractions. A tutor can work at your student's pace so that tutoring is efficient while using their learning style - so that tutoring is effective. To learn more about how tutoring can help your student master conditional statements, contact the Educational Directors at Varsity Tutors today.

• ASCP Board of Certification - American Society for Clinical Pathology Board of Certification Training
• Holistic Medicine Tutors
• Internal Medicine Tutors
• DAT Courses & Classes
• Haitian Creole Tutors
• North Carolina Bar Exam Courses & Classes
• Actuarial Exam PA Test Prep
• Spanish 3 Tutors
• SSAT Test Prep
• Livonian Tutors
• Computer Programming Tutors
• CISM - Certified Information Security Manager Courses & Classes
• ARM - Associate in Risk Management Courses & Classes
• Arizona Bar Exam Test Prep
• SHSAT Test Prep
• REGENTS Tutors
• Series 57 Tutors
• WEST-E Test Prep
• Honors Science Tutors
• High School Level American Literature Tutors
• Kansas City Tutoring
• Austin Tutoring
• San Francisco-Bay Area Tutoring
• Buffalo Tutoring
• Philadelphia Tutoring
• Louisville Tutoring
• Atlanta Tutoring
• Dayton Tutoring
• San Antonio Tutoring
• Albuquerque Tutoring
• GRE Tutors in Los Angeles
• ACT Tutors in Philadelphia
• LSAT Tutors in Washington DC
• English Tutors in Atlanta
• SSAT Tutors in Los Angeles
• Physics Tutors in Seattle
• French Tutors in Seattle
• SSAT Tutors in Dallas Fort Worth
• ISEE Tutors in Dallas Fort Worth
• English Tutors in San Diego

Conditional Statement

A conditional statement is a part of mathematical reasoning which is a critical skill that enables students to analyze a given hypothesis without any reference to a particular context or meaning. In layman words, when a scientific inquiry or statement is examined, the reasoning is not based on an individual's opinion. Derivations and proofs need a factual and scientific basis. 

Mathematical critical thinking and logical reasoning are important skills that are required to solve maths reasoning questions.

In this mini-lesson, we will explore the world of conditional statements. We will walk through the answers to the questions like what is meant by a conditional statement, what are the parts of a conditional statement, and how to create conditional statements along with solved examples and interactive questions.

Lesson Plan  

What is meant by a conditional statement.

A statement that is of the form "If p, then q" is a conditional statement. Here 'p' refers to 'hypothesis' and 'q' refers to 'conclusion'.

For example, "If Cliff is thirsty, then she drinks water."

This is a conditional statement. It is also called an implication.

'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B. 

Here are two more conditional statement examples

Example 1: If a number is divisible by 4, then it is divisible by 2.

Example 2: If today is Monday, then yesterday was Sunday.

What Are the Parts of a Conditional Statement?

Hypothesis (if) and Conclusion (then) are the two main parts that form a conditional statement.

Let us consider the above-stated example to understand the parts of a conditional statement.

Conditional Statement : If today is Monday, then yesterday was Sunday.

Hypothesis : "If today is Monday."

Conclusion : "Then yesterday was Sunday."

On interchanging the form of statement the relationship gets changed.

To check whether the statement is true or false here, we have subsequent parts of a conditional statement. They are:

• Contrapositive

Biconditional Statement

Let us consider hypothesis as statement A and Conclusion as statement B.

Following are the observations made:

Converse of Statement

When hypothesis and conclusion are switched or interchanged, it is termed as converse statement . For example,

Conditional Statement : “If today is Monday, then yesterday was Sunday.”

Hypothesis : “If today is Monday”

Converse : “If yesterday was Sunday, then today is Monday.”

Here the conditional statement logic is, If B, then A (B → A)

Inverse of Statement

When both the hypothesis and conclusion of the conditional statement are negative, it is termed as an inverse of the statement. For example,

Conditional Statement: “If today is Monday, then yesterday was Sunday”.

Inverse : “If today is not Monday, then yesterday was not Sunday.”

Here the conditional statement logic is, If not A, then not B (~A → ~B)

Contrapositive Statement

When the hypothesis and conclusion are negative and simultaneously interchanged, then the statement is contrapositive. For example,

Contrapositive: “If yesterday was not Sunday, then today is not Monday”

Here the conditional statement logic is, if not B, then not A (~B → ~A)

The statement is a biconditional statement when a statement satisfies both the conditions as true, being conditional and converse at the same time. For example,

Biconditional : “Today is Monday if and only if yesterday was Sunday.”

Here the conditional statement logic is, A if and only if B (A ↔ B)

How to Create Conditional Statements?

Here, the point to be kept in mind is that the 'If' and 'then' part must be true.

If a number is a perfect square , then it is even.

• 'If' part is a number that is a perfect square.

Think of 4 which is a perfect square.

This has become true.

• The 'then' part is that the number should be even. 4 is even.

This has also become true. 

Thus, we have set up a conditional statement.

Let us hypothetically consider two statements, statement A and statement B. Observe the truth table for the statements:

According to the table, only if the hypothesis (A) is true and the conclusion (B) is false then, A → B will be false, or else A → B will be true for all other conditions.

• A sentence needs to be either true or false, but not both, to be considered as a mathematically accepted statement.
• Any sentence which is either imperative or interrogative or exclamatory cannot be considered a mathematically validated statement. 
• A sentence containing one or many variables is termed as an open statement. An open statement can become a statement if the variables present in the sentence are replaced by definite values.

Solved Examples

Let us have a look at a few solved examples on conditional statements.

Identify the types of conditional statements.

There are four types of conditional statements:

• If condition
• If-else condition
• Nested if-else
• If-else ladder.

Ray tells "If the perimeter of a rectangle is 14, then its area is 10."

Which of the following could be the counterexamples? Justify your decision.

a) A rectangle with sides measuring 2 and 5

b) A rectangle with sides measuring 10 and 1

c) A rectangle with sides measuring 1 and 5

d) A rectangle with sides measuring 4 and 3

a) Rectangle with sides 2 and 5: Perimeter = 14 and area = 10

Both 'if' and 'then' are true.

b) Rectangle with sides 10 and 1: Perimeter = 22 and area = 10

'If' is false and 'then' is true.

c) Rectangle with sides 1 and 5: Perimeter = 12 and area = 5

Both 'if' and 'then' are false.

d) Rectangle with sides 4 and 3: Perimeter = 14 and area = 12

'If' is true and 'then' is false.

Joe examined the set of numbers {16, 27, 24} to check if they are the multiples of 3. He claimed that they are divisible by 9. Do you agree or disagree? Justify your answer.

Conditional statement : If a number is a multiple of 3, then it is divisible by 9.

Let us find whether the conditions are true or false.

a) 16 is not a multiple of 3. Thus, the condition is false. 

16 is not divisible by 9. Thus, the conclusion is false. 

b) 27 is a multiple of 3. Thus, the condition is true.

27 is divisible by 9. Thus, the conclusion is true. 

c) 24 is a multiple of 3. Thus the condition is true.

24 is not divisible by 9. Thus the conclusion is false.

Write the converse, inverse, and contrapositive statement for the following conditional statement. 

If you study well, then you will pass the exam.

The given statement is - If you study well, then you will pass the exam.

It is of the form, "If p, then q"

The converse statement is, "You will pass the exam if you study well" (if q, then p).

The inverse statement is, "If you do not study well then you will not pass the exam" (if not p, then not q).

The contrapositive statement is, "If you did not pass the exam, then you did not study well" (if not q, then not p).

Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

Let's Summarize

The mini-lesson targeted the fascinating concept of the conditional statement. The math journey around conditional statements started with what a student already knew and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever.

About Cuemath

At  Cuemath , our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

FAQs on Conditional Statement

1. what is the most common conditional statement.

'If and then' is the most commonly used conditional statement.

2. When do you use a conditional statement?

Conditional statements are used to justify the given condition or two statements as true or false.

3. What is if and if-else statement?

If is used when a specified condition is true. If-else is used when a particular specified condition is not satisfying and is false.

4. What is the symbol for a conditional statement?

'\(\rightarrow\)' is the symbol used to represent the relation between two statements. For example, A\(\rightarrow\)B. It is known as the logical connector. It can be read as A implies B.

5. What is the Contrapositive of a conditional statement?

If not B, then not A (~B → ~A)

6. What is a universal conditional statement?

Conditional statements are those statements where a hypothesis is followed by a conclusion. It is also known as an " If-then" statement. If the hypothesis is true and the conclusion is false, then the conditional statement is false. Likewise, if the hypothesis is false the whole statement is false. Conditional statements are also termed as implications.

Conditional Statement: If today is Monday, then yesterday was Sunday

Hypothesis: "If today is Monday."

Conclusion: "Then yesterday was Sunday."

If A, then B (A → B)

• Live one on one classroom and doubt clearing
• Practice worksheets in and after class for conceptual clarity
• Personalized curriculum to keep up with school

2.4 Truth Tables for the Conditional and Biconditional

Learning objectives.

After completing this section, you should be able to:

• Use and apply the conditional to construct a truth table.
• Use and apply the biconditional to construct a truth table.
• Use truth tables to determine the validity of conditional and biconditional statements.

Computer languages use if-then or if-then-else statements as decision statements:

• If the hypothesis is true, then do something.
• Or, if the hypothesis is true, then do something; else do something else.

For example, the following representation of computer code creates an if-then-else decision statement:

Check value of variable i i .

If i < 1 i < 1 , then print "Hello, World!" else print "Goodbye".

In this imaginary program, the if-then statement evaluates and acts on the value of the variable i i . For instance, if i = 0 i = 0 , the program would consider the statement i < 1 i < 1 as true and “Hello, World!” would appear on the computer screen. If instead, i = 3 i = 3 , the program would consider the statement i < 1 i < 1 as false (because 3 is greater than 1), and print “Goodbye” on the screen.

In this section, we will apply similar reasoning without the use of computer programs.

People in Mathematics

The Countess of Lovelace, Ada Lovelace, is credited with writing the first computer program. She wrote an algorithm to work with Charles Babbage’s Analytical Engine that could compute the Bernoulli numbers in 1843. In doing so, she became the first person to write a program for a machine that would produce more than just a simple calculation. The computer programming language ADA is named after her.

Reference: Posamentier, Alfred and Spreitzer Christian, “Chapter 34 Ada Lovelace: English (1815-1852)” pp. 272-278, Math Makers: The Lives and Works of 50 Famous Mathematicians , Prometheus Books, 2019.

Use and Apply the Conditional to Construct a Truth Table

A conditional is a logical statement of the form if p p , then q q . The conditional statement in logic is a promise or contract. The only time the conditional, p → q , p → q , is false is when the contract or promise is broken.

For example, consider the following scenario. A child’s parent says, “If you do your homework, then you can play your video games.” The child really wants to play their video games, so they get started right away, finish within an hour, and then show their parent the completed homework. The parent thanks the child for doing a great job on their homework and allows them to play video games. Both the parent and child are happy. The contract was satisfied; true implies true is true.

Now, suppose the child does not start their homework right away, and then struggles to complete it. They eventually finish and show it to their parent. The parent again thanks the child for completing their homework, but then informs the child that it is too late in the evening to play video games, and that they must begin to get ready for bed. Now, the child is really upset. They held up their part of the contract, but they did not receive the promised reward. The contract was broken; true implies false is false.

So, what happens if the child does not do their homework? In this case, the hypothesis is false. No contract has been entered, therefore, no contract can be broken. If the conclusion is false, the child does not get to play video games and might not be happy, but this outcome is expected because the child did not complete their end of the bargain. They did not complete their homework. False implies false is true. The last option is not as intuitive. If the parent lets the child play video games, even if they did not do their homework, neither parent nor child are going to be upset. False implies true is true.

The truth table for the conditional statement below summarizes these results.

Notice that the only time the conditional statement, p → q , p → q , is false is when the hypothesis, p p , is true and the conclusion, q q , is false.

Logic Part 8: The Conditional and Tautologies

Example 2.18

Constructing truth tables for conditional statements.

Assume both of the following statements are true: p p : My sibling washed the dishes, and q q : My parents paid them $5.00. Create a truth table to determine the truth value of each of the following conditional statements.

• p → q p → q
• p → ~ q p → ~ q
• ~ p → q ~ p → q

Your Turn 2.18

Example 2.19, determining validity of conditional statements.

Construct a truth table to analyze all possible outcomes for each of the following statements then determine whether they are valid.

• p ∧ q → ~ q p ∧ q → ~ q
• p → ~ p ∨ q p → ~ p ∨ q

Your Turn 2.19

Use and apply the biconditional to construct a truth table.

The biconditional, p ↔ q p ↔ q , is a two way contract; it is equivalent to the statement ( p → q ) ∧ ( q → p ) . ( p → q ) ∧ ( q → p ) . A biconditional statement, p ↔ q , p ↔ q , is true whenever the truth value of the hypothesis matches the truth value of the conclusion, otherwise it is false.

The truth table for the biconditional is summarized below.

Logic Part 11B: Biconditional and Summary of Truth Value Rules in Logic

Example 2.20

Constructing truth tables for biconditional statements.

Assume both of the following statements are true: p p : The plumber fixed the leak, and q q : The homeowner paid the plumber $150.00. Create a truth table to determine the truth value of each of the following biconditional statements.

• p ↔ q p ↔ q
• p ↔ ~ q p ↔ ~ q
• ~ p ↔ ~ q ~ p ↔ ~ q

Your Turn 2.20

The biconditional, p ↔ q , p ↔ q , is true whenever the truth values of p p and q q match, otherwise it is false.

Logic Part 13: Truth Tables to Determine if Argument is Valid or Invalid

Example 2.21

Determining validity of biconditional statements.

Construct a truth table to analyze all possible outcomes for each of the following statements, then determine whether they are valid.

• p ∧ q ↔ p ∧ ~ q p ∧ q ↔ p ∧ ~ q
• p ∨ q ↔ ~ p ∨ q p ∨ q ↔ ~ p ∨ q
• p → q ↔ ~ q → ~ p p → q ↔ ~ q → ~ p
• p ∧ q → ~ r ↔ p ∧ q ∧ r p ∧ q → ~ r ↔ p ∧ q ∧ r

Your Turn 2.21

Check your understanding, section 2.4 exercises.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/contemporary-mathematics/pages/1-introduction

• Authors: Donna Kirk
• Publisher/website: OpenStax
• Book title: Contemporary Mathematics
• Publication date: Mar 22, 2023
• Location: Houston, Texas
• Book URL: https://openstax.org/books/contemporary-mathematics/pages/1-introduction
• Section URL: https://openstax.org/books/contemporary-mathematics/pages/2-4-truth-tables-for-the-conditional-and-biconditional

© Dec 21, 2023 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Calcworkshop

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

Get access to all the courses and over 450 HD videos with your subscription

Monthly and Yearly Plans Available

Get My Subscription Now

Still wondering if CalcWorkshop is right for you? Take a Tour and find out how a membership can take the struggle out of learning math.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

2.1: Statements and Logical Operators

• Last updated
• Save as PDF
• Page ID 7039

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

PREVIEW ACTIVITY \(\PageIndex{1}\): Compound Statements

Mathematicians often develop ways to construct new mathematical objects from existing mathematical objects. It is possible to form new statements from existing statements by connecting the statements with words such as “and” and “or” or by negating the statement. A logical operator (or connective ) on mathematical statements is a word or combination of words that combines one or more mathematical statements to make a new mathematical statement. A compound statement is a statement that contains one or more operators. Because some operators are used so frequently in logic and mathematics, we give them names and use special symbols to represent them.

• The conjunction of the statements \(P\) and \(Q\) is the statement “\(P\) and \(Q\)” and its denoted by \(P \wedge Q\). The statement \(P \wedge Q\) is true only when both \(P\) and \(Q\) are true.
• The disjunction of the statements \(P\) and \(Q\) is the statement “\(P\) or \(Q\)” and its denoted by \(P \vee Q\). The statement \(P \vee Q\) is true only when at least one of \(P\) or \(Q\) is true.
• The negation ( of a statement ) of the statement \(P\) is the statement “ not \(P\) ” and is denoted by \(\urcorner P\). The negation of \(P\) is true only when \(P\) is false, and \(\urcorner P\) is false only when \(P\) is true.
• The implication or conditional is the statement “ If \(P\) then \(Q\)” and is denoted by \(P \to Q\). The statement \(P \to Q\) is often read as “\(P\) implies \(Q\), and we have seen in Section 1.1 that \(P \to Q\) is false only when \(P\) is true and \(Q\) is false.

Some comments about the disjunction. It is important to understand the use of the operator “or.” In mathematics, we use the “ inclusive or ” unless stated otherwise. This means that \(P \vee Q\) is true when both \(P\) and \(Q\) are true and also when only one of them is true. That is, \(P \vee Q\) is true when at least one of \(P\) or \(Q\) is true, or \(P \vee Q\) is false only when both \(P\) and \(Q\) are false.

A different use of the word “or” is the “ exclusive or .” For the exclusive or, the resulting statement is false when both statements are true. That is, “\(P\) exclusive or \(Q\)” is true only when exactly one of \(P\) or \(Q\) is true. In everyday life, we often use the exclusive or. When someone says, “At the intersection, turn left or go straight,” this person is using the exclusive or.

Some comments about the negation . Although the statement, \(\urcorner P\), can be read as “It is not the case that \(P\),” there are often betters ways to say or write this in English. For example, we would usually say (or write):

• The negation of the statement, “391 is prime” is “391 is not prime.”
• The negation of the statement, “\(12 < 9\)” is “\(12 \ge 9\).”

\(P\): 15 is odd \(Q\): 15 is prime write each of the following statements as English sentences and determine

whether they are true or false. (a) \(P \wedge Q\). (b) \(P \vee Q\). (c) \(P \wedge \urcorner Q\). (d) \(\urcorner P \vee \urcorner Q\).

P : 15 is odd R: 15 < 17

write each of the following statements in symbolic form using the operators\(\wedge\), \(\vee\), and \(\urcorner\)

(a) 15 \(\ge\) 17. (b) 15 is odd or 15 \(\ge\) 17. (c) 15 is even or 15 <17. (d) 15 is odd and 15 \(\ge\) 17.

PREVIEW ACTIVITY\(\PageIndex{2}\): Truth Values of Statements

We will use the following two statements for all of this Preview Activity:

• \(P\) is the statement “It is raining.”
• \(Q\) is the statement “Daisy is playing golf.”

In each of the following four parts, a truth value will be assigned to statements \(P\) and \(Q\). For example, in Question (1), we will assume that each statement is true. In Question (2), we will assume that \(P\) is true and \(Q\) is false. In each part, determine the truth value of each of the following statements:

(a) (\(P \wedge Q\)) It is raining and Daisy is playing golf.

(b) (\(P \vee Q\)) It is raining or Daisy is playing golf.

(c) (\(P \to Q\)) If it is raining, then Daisy is playing golf.

(d) (\(\urcorner P\)) It is not raining.

Which of the four statements [(a) through (d)] are true and which are false in each of the following four situations?

1. When \(P\) is true (it is raining) and \(Q\) is true (Daisy is playing golf). 2. When \(P\) is true (it is raining) and \(Q\) is false (Daisy is not playing golf). 3. When \(P\) is false (it is not raining) and \(Q\) is true (Daisy is playing golf). 4. When \(P\) is false (it is not raining) and \(Q\) is false (Daisy is not playing golf).

In the preview activities for this section, we learned about compound statements and their truth values. This information can be summarized with truth tables as is shown below.

Rather than memorizing the truth tables, for many people it is easier to remember the rules summarized in Table 2.1.

Other Forms of Conditional Statements

Conditional statements are extremely important in mathematics because almost all mathematical theorems are (or can be) stated in the form of a conditional statement in the following form:

If “certain conditions are met,” then “something happens.”

It is imperative that all students studying mathematics thoroughly understand the meaning of a conditional statement and the truth table for a conditional statement.

• If \(P\), then \(Q\).
• \(P\) implies \(Q\).
• \(P\) only if \(Q\).
• \(Q\) if \(P\).
• Whenever \(P\) is true, \(Q\) is true.
• \(Q\) is true whenever \(P\) is true.
• \(Q\) is necessary for \(P\). (This means that if \(P\) is true, then \(Q\) is necessarily true.)

In all of these cases, \(P\) is the hypothesis of the conditional statement and \(Q\) is the conclusion of the conditional statement.

Progress Check 2.1: The "Only if" statemenT

Recall that a quadrilateral is a four-sided polygon. Let \(S\) represent the following true conditional statement:

If a quadrilateral is a square, then it is a rectangle.

Write this conditional statement in English using

• the word “whenever”
• the phrase “only if”
• the phrase “is necessary for”
• the phrase “is sufficient for”

Add texts here. Do not delete this text first.

Constructing Truth Tables

Truth tables for compound statements can be constructed by using the truth tables for the basic connectives. To illustrate this, we will construct a truth table for. \((P \wedge \urcorner Q) \to R\). The first step is to determine the number of rows needed.

• For a truth table with two different simple statements, four rows are needed since there are four different combinations of truth values for the two statements. We should be consistent with how we set up the rows. The way we will do it in this text is to label the rows for the first statement with (T, T, F, F) and the rows for the second statement with (T, F, T, F). All truth tables in the text have this scheme.
• For a truth table with three different simple statements, eight rows are needed since there are eight different combinations of truth values for the three statements. Our standard scheme for this type of truth table is shown in Table 2.2 .

The next step is to determine the columns to be used. One way to do this is to work backward from the form of the given statement. For \((P \wedge \urcorner Q) \to R\), the last step is to deal with the conditional operator \((\to)\). To do this, we need to know the truth values of \((P \wedge \urcorner Q)\) and \(R\). To determine the truth values for \((P \wedge \urcorner Q)\), we need to apply the rules for the conjunction operator \((\wedge)\) and we need to know the truth values for \(P\) and \(\urcorner Q\).

Table 2.2 is a completed truth table for \((P \wedge \urcorner Q) \to R\) with the step numbers indicated at the bottom of each column. The step numbers correspond to the order in which the columns were completed.

• When completing the column for \(P \wedge \urcorner Q\), remember that the only time the conjunction is true is when both \(P\) and \(\urcorner Q\) are true.
• When completing the column for \((P \wedge \urcorner Q) \to R\), remember that the only time the conditional statement is false is when the hypothesis \((P \wedge \urcorner Q)\) is true and the conclusion, \(R\), is false.

The last column entered is the truth table for the statement \((P \wedge \urcorner Q) \to R\) using the set up in the first three columns.

Progress Check 2.2: Constructing Truth Tables

Construct a truth table for each of the following statements:

• \(P \wedge \urcorner Q\)
• \(\urcorner(P \wedge Q)\)
• \(\urcorner P \wedge \urcorner Q\)
• \(\urcorner P \vee \urcorner Q\)

Do any of these statements have the same truth table?

The Biconditional Statement

Some mathematical results are stated in the form “\(P\) if and only if \(Q\)” or “\(P\) is necessary and sufficient for \(Q\).” An example would be, “A triangle is equilateral if and only if its three interior angles are congruent.” The symbolic form for the biconditional statement “\(P\) if and only if \(Q\)” is \(P \leftrightarrow Q\). In order to determine a truth table for a biconditional statement, it is instructive to look carefully at the form of the phrase “\(P\) if and only if \(Q\).” The word “and” suggests that this statement is a conjunction. Actually it is a conjunction of the statements “\(P\) if \(Q\)” and “\(P\) only if \(Q\).” The symbolic form of this conjunction is \([(Q \to P) \wedge (P \to Q]\).

Progress Check 2.3: The Truth Table for the Biconditional Statement

Complete a truth table for \([(Q \to P) \wedge (P \to Q]\). Use the following columns: \(P\), \(Q\), \(Q \to P\), \(P \to Q\), and \([(Q \to P) \wedge (P \to Q]\). The last column of this table will be the truth for \(P \leftrightarrow Q\).

Other Forms of the Biconditional Statement

As with the conditional statement, there are some common ways to express the biconditional statement, \(P \leftrightarrow Q\), in the English language.

• \(P\) is and only if \(Q\).
• \(P\) is necessary and sufficient for \(Q\).
• \(P\) implies \(Q\) and \(Q\) implies \(P\).

Tautologies and Contradictions

Definition: tautology

A tautology is a compound statement S that is true for all possible combinations of truth values of the component statements that are part of \(S\). A contradiction is a compound statement that is false for all possible combinations of truth values of the component statements that are part of \(S\).

That is, a tautology is necessarily true in all circumstances, and a contradiction is necessarily false in all circumstances.

Progress Check 2.4 (Tautologies and Contradictions)

For statements \(P\) and \(Q\):

• Use a truth table to show that \((P \vee \urcorner P)\) is a tautology.
• Use a truth table to show that \((P \wedge \urcorner P)\) is a contradiction.
• Use a truth table to determine if \(P \to (P \vee P)\) is a tautology, a contradiction, nor neither.

Exercises for Section 2.1

• Suppose that Daisy says, “If it does not rain, then I will play golf.” Later in the day you come to know that it did rain but Daisy still played golf. Was Daisy’s statement true or false? Support your conclusion.
• Suppose that \(P\) and \(Q\) are statements for which \(P \to Q\) is true and for which \(\urcorner Q\) is true. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(P\) (b) \(P \wedge Q\) (c) \(P \vee Q\)
• Suppose that \(P\) and \(Q\) are statements for which \(P \to Q\) is false. What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\urcorner P \to Q\) (b) \(Q \to P\) (c) \(P \ vee Q\)
• Suppose that \(P\) and \(Q\) are statements for which \(Q\) is false and \(\urcorner P \to Q\) is true (and it is not known if \(R\) is true or false). What conclusion (if any) can be made about the truth value of each of the following statements? (a) \(\urcorner Q \to P\) (b) \(P\) (c) \(P \wedge R\) (d) \(R \to \urcorner P\)
• Construct a truth table for each of the following statements: (a) \(P \to Q\) (b) \(Q \to P\) (c) \(\urcorner P \to \urcorner Q\) (d) \(\urcorner Q \to \urcorner P\) Do any of these statements have the same truth table?
• Construct a truth table for each of the following statements: (a) \(P \vee \urcorner Q\) (b) \(\urcorner (P \vee Q)\) (c) \(\urcorner P \vee \urcorner Q\) (d) \(\urcorner P \wedge \urcorner Q\) Do any of these statements have the same truth table?
• Construct truth table for \(P \wedge (Q \vee R)\) and \((P \wedge Q) \vee (P \wedge R)\). What do you observe.
• Laura is in the seventh grade.
• ��Laura got an A on the mathematics test or Sarah got an A on the mathematics test.
• ��If Sarah got an A on the mathematics test, then Laura is not in the seventh grade. If possible, determine the truth value of each of the following statements. Carefully explain your reasoning. (a) Laura got an A on the mathematics test. (b) Sarah got an A on the mathematics test. (c) Either Laura or Sarah did not get an A on the mathematics test.
• Let \(P\) stand for “the integer \(x\) is even,” and let \(Q\) stand for “\(x^2\) is even.” Express the conditional statement \(P \to Q\) in English using (a) The "if then" form of the conditional statement (b) The word "Implies" (c) The "only if" form of the conditional statement (d) The phrase "is necessary for" (e) The phrase "is sufficient for"
• Repeat Exercise (9) for the conditional statement \(Q \to P\).
• For statements \(P\) and \(Q\), use truth tables to determine if each of the following statements is a tautology, a contradiction, or neither. (a) \(\urcorner Q \vee (P \to Q)\). (b) \(Q \wedge (P \wedge \urcorner Q)\). (c) \((Q \wedge P) \wedge (P \to \urcorner Q)\). (d) \(\urcorner Q \to (P \wedge \urcorner P)\).
• For statements \(P\), \(Q\), and \(R\): (a) Show that \([(P \to Q) \wedge P] \to Q\) is a tautology. Note : In symbolic logic, this is an important logical argument form called modus ponens . (b) Show that \([(P \to Q) \wedge (Q \to R)] \to (P \to R)\) is atautology. Note : In symbolic logic, this is an important logical argument form called syllogism . Explorations and Activities
• Working with Truth Values of Statements. Suppose that \(P\) and \(Q\) are true statements, that \(U\) and \(V\) are false statements, and that \(W\) is a statement and it is not known if \(W\) is true or false. Which of the following statements are true, which are false, and for which statements is it not possible to determine if it is true or false? Justify your conclusions. (a) \((P \vee Q) \vee (U \wedge W)\) (f) \((\urcorner P \vee \urcorner U) \wedge (Q \vee \urcorner V)\) (b) \(P \wedge (Q \to W)\) (g) \((P \wedge \urcorner Q) \wedge (U \vee W)\) (c) \(P \wedge (W \to Q)\) (h) \((P \vee \urcorner Q) \to (U \wedge W)\) (d) \(W \to (P \wedge U)\) (i) \((P \vee W) \to (U \wedge W)\) (e) \(W \to (P \wedge \urcorner U)\) (j) \((U \wedge \urcorner V) \to (P \wedge W)\)