Bijective Function: Overview, Questions, Preparation

First, we need to know what an injective and surjective function is. 

The injective function is a function that always links the distinct element of its domain to the distinctive element of its co-domain. It is a one-to-one function.

The surjective function is a function that maps one or more elements of A to the same element of B and is also called onto function. This can have more than one element in the co-domain. 

The Bijective Function is a combination of both: It is a one-to-one function that works onto the co-domain element.

How to Recognise a Bijective Function?

To recognize the bijection between two terms, we have first to understand the sets, map f: A → B, and conclude that |A| = |B|. To prove a function f is bijective, we obtain either the inverse or prove that it is both injective and surjective.

If two sets A and B are not of the same size, then the functions aren’t bijective because bijection is pairing up of the elements in the two sets perfectly. If |A| = |B| = n, then there exists n! bijections between A and B.

Example: Show that the function f(x) = 4x – 5 is a bijective function from R to R.
Given, f(x) = 4x – 5

The given function should be both injective and surjective.

(i) To Prove: The function is injective

For this f(a)=c and f(b)=c then a=b must be proved.
Let us take, f(a)=c and f(b)=c
Therefore, it can be written as:
c = 4a-5 and c = 4b-5
Thus, it can be written as: 4a-5 = 4b -5
Simplify the equation; we will get: a = b; hence the given function is injective

(ii) To Prove: The function is surjective
First, we should show that for “a” in R there exists a point “b” in the domain such that f(b) =a
Let, a = 4x -5
Therefore, b must be (a+5)/4

Since this is in R, the function is surjective.

As the function is both injective and surjective, it is bijective.

Class 12 students learn this in Relations and Functions, which has a 40% weightage in the board exams. This topic comes in the functions category and 2-4 mark questions. Here, students learn to categorise data in domains and understand whether a function is surjective or injective. It has lots of weightage in national entrance exams.

1. Is the mapping injective or surjective?
(i) { (x, y): x is a person, y is the father of x }.

Solution.

(i) Here, y can be a father to two terms in the x domain as it is not specific. Hence, this is a surjective function and not injective.

2. Prove that the function f: A → b is invertible only if f is both one-one and onto.

Solution.

A function f :X → Y is defined to be invertible, if there exists a function g = Y → X such that g(f) = I_x and f(g) = I_y, the function is called the inverse of f and is denoted by f^-1. This, if bijective, means f can be invertible.

3. If set a contains five elements, and the set b contains six elements, then the number of bijective mappings from a to b is

Solution.

0. Since the number of elements in B is more than A.

Q: Every function is invertible. True or false?

Q: Can a function be bijective if it is one-one only?

Q: Give an example of a bijective function?

Q: Give an example of a surjective function?

Q: What is a bijective function?