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Bijective Function (Bijection)

Bijective function connects elements of two sets such that, it is both one-one and onto function. The elements of the two sets are mapped in such a manner that every element of the range is in co-domain, and is related to a distinct domain element. In simple words, we can say that a function f: A→B is said to be a bijective function or bijection if f is both one-one (injective) and onto (surjective).

In this article, we will explore the concept of the bijective function, and define the concept, its conditions, its properties, and applications with the help of a diagram. We will go through various examples based on bijection to better understand the concept.

1. What Is a Bijective Function?
2. Bijective Function Definition
3. Bijective Function Conditions
4. Properties of Bijection
5. Injective Surjective Bijective
6. One-to-one Correspondence
7. FAQs on Bijective Function

What Is a Bijective Function?

A bijective function is a combination of an injective function and a surjective function. Bijective function relates elements of two sets A and B with the domain in set A and the co-domain in set B, such that every element in A is related to a distinct element in B, and every element of set B is the image of some element of set A.

The bijective function is both a one-one function and onto function. A bijective function from set A to set B has an inverse function from set B to set A. A bijective function of a set of elements defined to itself is called a permutation. Here every element of the set is related to itself.

Bijective Function

From the above examples of bijective function, we can observe that every element of set B has been related to a distinct element of set A. The non-bijective functions have some element in set B which do not have a pre-image in set A, or some of the elements in set B is the image for more than one element in set A.

Bijective Function Definition

A function f: A→B is said to be a bijective function if f is both one-one and onto, that is, every element in A has a unique image in B and every element of B has a pre-image in set A. In simple words, we can say that a function f is a bijection if it is both injection and surjection.

Bijective Function Conditions

There are some conditions that need to be satisfied for a function to be a bijection. The bijective functions need to satisfy the following four conditions.

  • Every element of set A must be paired with an element of set B.
  • The element of set A must not be paired with more than one element of set B.
  • Each element of set B must be paired with an element of set A.
  • The element of set B must not be paired with more than one element of set A.

Properties of Bijection

Now that we have understood the meaning of bijection, given below are a few important properties of bijective functions which are useful in understanding the concept better:

  • The domain and co-domain sets of a bijective function have an equal number of elements.
  • The codomain and range of the bijective function are the same.
  • The bijective function has an inverse function.
  • The inverse of a bijective function is also a bijection.
  • The bijection cannot be a constant function.
  • Bijective functions if represented as a graph is always a straight line.
  • The bijective function follows a reflexive, symmetric, and transitive property.
  • The composition of bijections f and g is also a bijective function. If f and g are bijective functions, then f o g is also a bijection.

Injective Surjective Bijective

In this section, we will discuss the meaning and differences between injective, surjective, and bijective functions. The injective function is also known as the one-one function, and the surjective function is also called the onto function.

Injective Surjective Bijective
A function f: X→Y is said to be injective when for each x1, x2 ∈ X if f(x1) = f(x2) then x1 = x2. A function f: X→Y is said to be surjective when, if for each y ∈ Y there exists some x in X such that f(x) = y. A function f: X→Y is said to be bijective if f is both one-one and onto.
Example: f: R→R defined as f(x) = 2x Example: For A = {1,−1,2,3} and B = {1,4,9}, f: A→B defined as f(x) = x2 is surjective. Example: Example: For A = {−1,2,3} and B = {1,4,9}, f: A→B defined as f(x) = x2 is bijective.

One-to-one Correspondence

One-to-One functions define that each element of one set called Set (A) is mapped with a unique element of another set called Set (B). A function f : X → Y is said to be one to one correspondence, if the images of unique elements of X under f are unique, i.e., for every x1 , x2 ∈ X, f(x1) = f(x2) implies x1  = x2 and also range = codomain. Otherwise, we call it a non-invertible function or not a bijective function. Therefore we can say, every element of the codomain of one-to-one correspondence is the image of only one element of its domain.

Important Notes on Bijective Function

  • A function is a bijection if it is both injective and surjective.
  • Every element in A has a unique image in the codomain and every element of the codomain has a pre-image in the domain.

Related Topics

 

Bijective Function Examples

  1. Example 1: Prove that the one-one function f : {1, 2, 3} → {4, 5, 6} is a bijective function.

    Solution:

    The given function f: {1, 2, 3} → {4, 5, 6} is a one-one function, and hence it relates every element in the domain to a distinct element in the co-domain set. The three elements of the domain set relate to all the three elements of the co-domain set. Also since the co-domain includes all the elements of the second set, the given function is also an onto function as the range is equal to the codomain.

    Therefore, the given function is a bijective function.

  2. Example 2: Show that the function f : N → N, given by f(x) = x + 1, if x is odd, and x - 1, if x is even, is a surjective function,

    Solution:

    Let f(x1) = f(x2). Further, let us suppose that x1 is odd, and x2 is even, then we have x1 + 1 = x2 - 1, or x2 - x1 = 2, which is not possible.

    Also the possibility of x1 being even, and x2 being odd is also ruled out, using the same argument. Therefore x1, and x2 both should be either odd or even.

    Let us assume both x1, and x2 to be odd, and we have f(x1) = f(x2) ⇒ x1 + 1 = x2 + 1 ⇒ x1 = x2. Also, if both x1, and x2 are even, we have \(f(x_1) = f(x_2)\) ) ⇒ f(x1) = f(x2) ⇒ x1 - 1 = x2 - 1 ⇒ x1 = x2. Therefore, the function f is a one-one function.

    Further, any odd number 2n + 1 in the codomain of N is the image of 2n + 2 in the domain of N, and any even number 2n in the co-domain of N, is the image of 2n - 1 in the domain N. Hence the function is onto function.

    Therefore, the given function is a bijective function.

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FAQs on Bijective Function

What Is a Bijective Function?

Bijective function relates elements of two sets such that every element of the domain set is related to a distinct element of the codomain set, and every element of the codomain set has been utilized.

How Do We Know If a Function Is a Bijective Function?

A function can be easily identified as a bijective function if it is a one-one function, and every element of the codomain set has a preimage in the domain set.

Are all Functions Bijections?

All the functions are not bijective functions. Some functions can only be injective, or only surjective functions. Some elements of the codomain set may not be utilized or the elements of the codomain set may be related to more than one element of the domain set.

How Many Types of Bijective Functions Are There?

There is only one bijective function, and it does not have any more classifications. A bijective function is a combination of an injective function and a surjective function.

Is x2 a Bijective Function?

A square function f(x) = x2 is not a bijective function if the codomain of the function is all real numbers.

Is the Inverse of a Bijective Function Bijective?

Yes, the inverse of a bijective function is also a bijective function. If a function f: A → B is defined as f(a) = b is bijective, then its inverse f-1(y) = x is also a bijection.

How to Prove Bijective Function?

To prove that a function is a bijective function, we need to show that every element of the domain has a unique image in the codomain set and each codomain element has a pre-image in the domain set.

Why do only Bijective Functions Have Inverses?

A function f: A→B has an inverse if and only if it is bijective so that every element of the codomain can be mapped back to an element of the domain that becomes the codomain of the inverse function.

What is One-to-one Correspondence?

A function f : X → Y is said to be one to one correspondence, if the images of unique elements of X under f are unique, i.e., for every x1 , x2 ∈ X, f(x1 ) = f(x2 ) implies x1  = x2 and also range = codomain.

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