FUNCTIONS

SUBMITTED BY:-Ms.MEHAK KAPOOR

PG DEPT COMP SC.&IT

Definition of Functions�

• Functions are an important part of discrete mathematics.
• 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.

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• The function can be represented as

f: A ⇢ B.

A is called the domain of the function

B is called the codomain function.

NOTE:- 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

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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)
• Many-One function
• Onto function( or Surjective Function)
• Into Function
• One-One Into function
• Many-one onto function
• Many-one into a 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.

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f: A ⇢ B is one-one

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

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

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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.

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

f: A ⇢ B is many-one

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

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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.

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• 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.

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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.

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

EXAMPLE

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.

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Many-one into a 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.

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• 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

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• (gof)(x) = g(f(x)), for all x ∈ A

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• is called the composition of f and g.

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