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.

What is Function?

• 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

Pre-Image and Image of a Function

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.

Domain of a Function

• 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.

Codomain (Range) of a Function

• 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.

Types 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

Many-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

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.

Into 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.

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:

• one-one i.e. f(a) = f(b)
  • ⇒ a = b for all a, b ∈  A
• onto i.e.  for each b ∈ B, there exists
  • a ∈ A such that f(a) = b.

One-One Into Function

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

Many-One Onto Function

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

Many-One Into Function

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

Inverse of a 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.

Composition of Functions

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.

Summary: Functions in Discrete Mathematics

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 = ⁿP_m
• Total number of onto functions = n^m - ⁿC₁(n-1)^m + ⁿC₂(n-2)^m - . . . + (-1)^n-1nC_n-11^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^-1o f^-1
• f^-1o f = f^-1(f(a)) = f^-1(b) = a
• fof^-1 = f(f^-1(b)) = f(a) = b

Sample Questions on Functions in Discrete Mathematics

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

Solution:

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) = x³ . Find fog and gof.

Solution:

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 = cos³x

fog (x) = f(g(x)) = f(x³) = cos x³

Question 3: If f : Q ⇢ Q is given by f(x) = x² , then find f^-1(16).

Solution:

Let f^-1(16) = x

f(x) = 16

⇒ x² = 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.

Solution:

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.

Solution:

Total number of functions = 3⁵ = 243

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