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Functions in Discrete Mathematics

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  • Last Updated : 27 May, 2022

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

Functions:

  • 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

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 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.

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.

Types of function:

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

ONE-ONE FUNCTION

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

MANY-ONE FUNCTION

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

ONTO FUNCTION

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.

INTO FUNCTION

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:

  • 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 CORRESPONDENT FUNCTION

ONE-ONE CORRESPONDENT FUNCTION

One-One Into function:

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

ONE-ONE INTO FUNCTION

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 ONTO FUNCTION

MANY-ONE ONTO FUNCTION

Many-one into a function:

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

MANY-ONE INTO FUNCTION

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

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 

is called the composition of f and g.

Note:

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 = nm
  • Total number of one-one function = nPm
  • Total number of onto functions = nmnC1(n-1)m + nC2(n-2)m – ………….. + (-1)n-1nCn-11m    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:

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

Sol:   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).

Ques 2: Let  f : R ⇢ R ; f(x) = cos x and  g : R ⇢ R ; g(x) = x3 . Find fog and gof.

Sol: 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 

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

Sol: 

Let f-1(16) = x
f(x) = 16
x2 = 16
x = ± 4 
f-1(16) = {-4, 4}

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

Sol: 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.

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

Sol: Total number of functions = 35 = 243


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