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Use a membership table to show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C). 

What is the symbolic word if Caroline is late (l), we will not pursue our plan (~p). We did not pursue our plan. Therefore, Caroline was late.

Show that the square of an even number is an even number using a direct proof.

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Patrick has assignments in 5 subjects. He can only do two assignments . In how many ways can do two assignments?

Patrick has assignments in 5 subjects he can only do two assignments in how many ways can he do two assignments

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Use proof by contradiction to show that a number is even if its square is even. Before showing your solution, rewrite the statement to the proper form of a conditional statement. Assign variables to the simple propositions then write your assumption using these variables and logic symbols. At the end of your solution

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Functions in Discrete Mathematics

Functions are an important part of discrete mathematics. This article is all about functions, their types, and other details of functions. A function assigns exactly one element of a set to each element of the other set. Functions are the rules that assign one input to one output. The function can be represented as f: A ⇢ B. A is called the domain of the function and B is called the codomain function.

Table of Content

What is Function?

Pre-image and image of a function, domain of a function, codomain (range) of a function, types of function, inverse of a function, composition of functions, summary: functions in discrete mathematics, sample questions on functions in discrete mathematics.

• A function assigns exactly one element of one set to each element of other sets.
• A function is a rule that assigns each input exactly one output.
• A function f from A to B is an assignment of exactly one element of B to each element of A (where  A and B are non-empty sets).
• A function f from set A to set B is represented as f: A ⇢ B where A is called the domain of f and B is called as codomain of f.
• If b is a unique element of B to element a of A assigned by function F then, it is written as f(a) = b.
• Function f maps A to B means f is a function from A to B i.e. f: A ⇢ B

A function f: A ⇢ B such that for each a ∈ A, there exists a unique b ∈ B such that (a, b) ∈ R then, a is called the pre-image of f and b is called the image of f.

• If f is a function from set A to set B then, A is called the domain of function f.
• The set of all inputs for a function is called its domain.
• If f is a function from set A to set B then, B is called the codomain of function f.
• The set of all allowable outputs for a function is called its codomain.

Read More about Domain and Range of Function .

Some of the common types of functions are:

• One-One Function

Many-One Function

• Onto Function

Into Function

• One-One Correspondent Function

One-One Into Function

Many-one onto function, many-one into function.

Let’s discuss these in detail.

One-One Function ( or Injective Function):

A function in which one element of the domain is connected to one element of the codomain.

A function f: A ⇢ B is said to be a one-one (injective) function if different elements of A have different images in B.

f: A ⇢ B is one-one 

⇒  a ≠ b ⇒  f(a) ≠ f(b)           for all a, b ∈  A

⇒  f(a) = f(b) ⇒ a = b           for all a, b ∈  A

ONE-ONE FUNCTION

A function f: A ⇢ B is said to be a many-one function if two or more elements of set A have the same image in B.

A function f: A ⇢ B is a many-one function if it is not a one-one function.

f: A ⇢ B is many-one.

⇒ a ≠ b but f(a) = f(b)      for all a, b ∈  A

MANY-ONE FUNCTION

Onto Function( or Surjective Function)

A function f: A ⇢ B is said to be onto (surjective) function if every element of B is an image of some element of A  i.e. f(A) = B or range of f is the codomain of f.

A function in which every element of the codomain has one pre-image.

  f: A ⇢ B is onto if for each b∈ B, there exists a∈ A such that f(a) = b.

ONTO FUNCTION

A function f: A ⇢ B is said to be an into a function if there exists an element in B with no pre-image in A.

A function f: A ⇢ B is into function when it is not onto.

INTO FUNCTION

One-One Correspondent Function (or Bijective Function or One-One Onto Function)

A function which is both one-one and onto (both injective and surjective) is called one-one correspondent(bijective) function. 

f : A ⇢ B is one-one correspondent (bijective) if:

• ⇒ a = b for all a, b ∈  A
• a ∈ A such that f(a) = b.

ONE-ONE CORRESPONDENT FUNCTION

A function that is both one-one and into is called one-one into function.

ONE-ONE INTO FUNCTION

A function that is both many-one and onto is called many-one onto function.

MANY-ONE ONTO FUNCTION

A function that is both many-one and into is called many-one into function.

MANY-ONE INTO FUNCTION

Let f: A ⇢ B be a bijection then, a function g: B ⇢ A which associates each element b ∈ B to a different element a ∈ A such that f(a) = b is called the inverse of f.

f(a) = b  ↔︎  g(b) = a

Read More about Inverse Function .

Let f: A ⇢ B and g: B ⇢ C be two functions then, a function gof: A ⇢ C is defined by 

(gof)(x) = g(f(x)), for all x ∈ A 

Here, (gof)(x) is called the composition of f and g.

Read More about Composition of Function .

Let X and Y be two sets with m and n elements and a function is defined as f : X⇢Y then,

• Total number of functions = n m
• Total number of one-one function = n P m
• Total number of onto functions = n m – n C 1 (n-1) m + n C 2 (n-2) m – . . . + (-1) n-1n C n-1 1 m    if m ≥ n.

For the composition of functions f and g be two functions : 

• fog ≠ gof 
• If f and g both are one-one function then fog is also one-one.
• If f and g both are onto function then fog is also onto.
• If f and fog both are one-one function then g is also one-one.
• If f and fog both are onto function then it is not necessary that g is also onto.
• (fog) -1 = g -1 o f -1
• f -1 o f = f -1 (f(a)) = f -1 (b) = a
• fof -1 = f(f -1 (b)) = f(a) = b

Question 1: Show that the function f : R ⇢ R, given by f(x) = 2x, is one-one and onto.

For one-one: Let a, b ∈ R such that f(a) = f(b) then, f(a) = f(b) ⇒ 2a = 2b ⇒ a = b Therefore, f: R ⇢ R is one-one. For onto: Let p be any real number in R (co-domain). f(x) = p ⇒ 2x = p ⇒ x = p/2 p/2 ∈ R for p ∈ R such that f(p/2) = 2(p/2) = p For each p∈ R (codomain) there exists x = p/2 ∈ R (domain) such that f(x) = y For each element in codomain has its pre-image in domain. So, f: R ⇢ R is onto. Since f: R ⇢ R is both one-one and onto. f : R ⇢ R is one-one correspondent (bijective function).

Question 2: Let  f : R ⇢ R ; f(x) = cos x and  g : R ⇢ R ; g(x) = x 3 . Find fog and gof.

Since the range of f is a subset of the domain of g and the range of g is a subset of the domain of f. So, fog and gof both exist.  gof (x) = g(f(x)) = g(cos x) = (cos x) 3 = cos 3 x fog (x) = f(g(x)) = f(x 3 ) = cos x 3

Question 3: If f : Q ⇢ Q is given by f(x) = x 2 , then find f -1 (16).

Let f -1 (16) = x f(x) = 16 ⇒ x 2 = 16 ⇒ x = ± 4 Thus, f -1 (16) = {-4, 4}

Question 4: If f : R ⇢ R; f(x) = 2x + 7  is a bijective function then, find the inverse of f.

Let x ∈ R (domain), y ∈ R (codomain) such that f(a) = b f(x) = y ⇒ 2x + 7 = y ⇒ x = (y -7)/2 ⇒ f -1 (y) = (y -7)/2 Thus, f -1 : R ⇢ R is defined as f -1(x) = (x -7)/2 for all x∈ R.

Question 5: If f : A ⇢ B and |A| = 5 and |B| = 3 then find total number of functions.

Total number of functions = 3 5 = 243

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FAQs on Functions in Discrete Mathematics

What is a function in discrete mathematics.

A function in discrete mathematics is a relation between a set of inputs (domain) and a set of possible outputs (range), such that each input is related to exactly one output.

How is a function represented in mathematics?

A function is typically represented using function notation, such as f(x) or g(y), where f and g are the names of the functions and x and y are the input variables.

What is the difference between a function and a relation?

A function is a special type of relation where each input is related to exactly one output. In a relation, an input can be related to more than one output.

What are the different types of functions?

There are several types of functions, including: One-to-one functions: Each input is related to a unique output. Onto functions: Every element in the output set is related to by at least one element in the input set. Many-to-one functions: Multiple inputs can be related to the same output.

How do you determine if a function is injective, surjective, or bijective?

Injective (one-to-one): If each element in the input set is related to a unique element in the output set. Surjective (onto): If every element in the output set is related to by at least one element in the input set. Bijective: If a function is both injective and surjective, meaning each input is related to a unique output and every output is related to by at least one input.

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