Bijective Functions: FAQs

1. Define Bijective Function.

A bijective function, also known as a bijection or one-to-one function, is a function that connects two sets, Set A and Set B. In this function, every element from Set A points to a distinct element in Set B and it covers the entire Set B.

2. How Do You Determine If a Function Is Bijective?

To determine if a function is bijective, we need to check two things:

• Injectivity: This means that no two different items from Set A map to the same item in Set B through the function.
• Surjectivity: This ensures that every item in Set B has a corresponding item in Set A through the function.

3. What are the Two Conditions for a Function to be Bijective?

For a function to be bijective, it must satisfy two conditions:

• It must be injective, meaning that no two different items from Set A map to the same item in Set B.
• It must be surjective, indicating that every item in Set B has a corresponding item in Set A.

4. What Is the Difference Between Injective, Surjective, and Bijective Functions?

Injective functions (one-to-one) have unique image for each element of the domain. Surjective functions (onto function) cover the entire codomain. Bijective functions are both injective and surjective, establishing a one-to-one correspondence.

5. Can All Functions Be Bijective?

No, not all functions can be bijective. A function can only be bijective if it is both injective (one-to-one) and surjective (onto). Many functions are not injective or surjective.

6. What is One to One Correspondence?

One-to-one correspondence, or a bijection, is a relationship between two sets where each element in one set is paired with exactly one element in the other set, without duplication or omission.

7. Can a Finite Set Have a Bijective Function with an Infinite Set?

No, a finite set cannot have a bijective function with an infinite set. A bijective function implies a one-to-one correspondence, and it’s not possible between finite and infinite sets.

8. Are Bijections Unique Between Two Sets?

No, bijections between two sets are not unique. Different bijections can exist between the same two sets, but they all establish a one-to-one correspondence.

Bijective Function is a special type of function that represents the relationship between two sets in such a way that all elements in the domain have an image in the codomain and each element in the codomain has a pre-image in the domain.

Bijective Function is also called one-to-one correspondence due to the relationship between domain and codomain i.e., each element of the domain is mapped to a unique element of codomain, and no element of codomain remains left without pre-image. In real life, the concept of bijective functions is used in various places such as shuffling of cards, cryptography, biometrics, physical lock and keys, language translation, etc. In this article, we have provided a well-detailed description of the concept of the Bijective Function, including all the subtopics.