Notes On Bijective Functions - CBSE Class 12 Maths

A function, f : A B, is defined to be one-one, if every element of A is mapped to a unique element of B.

A function, f : A B, is said to be onto, if for every element y of B , there is an element x in A such that f(x) = y.

The function, f: A → B, is one-one and onto then that function, f: A → B, is a bijective function or a bijection.

A function,f: A → B, is said to be a bijection if it is both one-one and onto.

Ex:

X = {a, b, c}

Define a function, f: X → X, such that it is one-one.

Range of f = {a, b, c}

Range of f = Codomain of f

If the codomain and the range of a function are equal, then the function is onto.

Hence, the one-one mapping, f: X → X, is also an onto mapping.

A one-one function f : X → X is necessarily onto, for every finite set X.

An onto function f : X → X is necessarily one-one, for every finite set X.

X = {a, b, c}

Define a function, f : X → X, such that it is onto.

Range of f = Codomain of f

To show: f is one-one.

Assume that f is not one-one.

Then, two elements from the domain of X are mapped to the same element in the codomain of f.

Range of f < Codomain of f

Hence, an onto function defined from a finite set to itself is always one-one.

Summary

A function, f : A B, is defined to be one-one, if every element of A is mapped to a unique element of B.

A function, f : A B, is said to be onto, if for every element y of B , there is an element x in A such that f(x) = y.

The function, f: A → B, is one-one and onto then that function, f: A → B, is a bijective function or a bijection.

A function,f: A → B, is said to be a bijection if it is both one-one and onto.

Ex:

X = {a, b, c}

Define a function, f: X → X, such that it is one-one.

Range of f = {a, b, c}

Range of f = Codomain of f

If the codomain and the range of a function are equal, then the function is onto.

Hence, the one-one mapping, f: X → X, is also an onto mapping.

A one-one function f : X → X is necessarily onto, for every finite set X.

An onto function f : X → X is necessarily one-one, for every finite set X.

X = {a, b, c}

Define a function, f : X → X, such that it is onto.

Range of f = Codomain of f

To show: f is one-one.

Assume that f is not one-one.

Then, two elements from the domain of X are mapped to the same element in the codomain of f.

Range of f < Codomain of f

Hence, an onto function defined from a finite set to itself is always one-one.

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