-Bijective Function

Definition

A Bijective Function, also known as an Injective function or One-to-One function, is a function between two sets that maps every Element of the Domain to exactly one Element of the Codomain. In other words, it is a function that assigns each Element in the Domain to exactly one Element in the range.

Definition of Key Terms

Properties of Bijective Functions

Bijective functions have several important properties:

1. Every Element in the Codomain Is Attained Exactly Once

By definition, every Element in the Codomain is mapped to exactly one Element by the function. This means that there are no Duplicate elements in the Codomain.

2. The Function Is One-to-One

As mentioned earlier, a Bijective Function is also known as an Injective function or One-to-One function. This property ensures that each Element in the Domain maps to a Unique Element in the Codomain.

Example of a Bijective Function

Let’s consider a simple example:

Domain Codomain
{1, 2} {a, b}

The function f from the Set {1, 2} to the Set {a, b} is Bijective because every Element in the Domain maps to exactly one Element in the Codomain. Specifically:

  • f(1) = a
  • f(2) = b

Counterexample

A Counterexample of a non-Bijective Function is the Identity Function id from the Set {1, 2} to itself:

Domain Codomain
{1, 2} {1, 2}

The function id maps every Element in the Domain to exactly one Element in the Codomain. However, it is not Bijective because there are multiple elements in the Codomain (both a and b). Specifically:

  • id(1) = 1
  • id(2) = 2

Conclusion

In conclusion, a Bijective Function is a function that maps every Element in the Domain to exactly one Element in the Codomain. It has several important properties, including every Element in the Codomain being attained exactly once and the function being One-to-One. A simple example of a Bijective Function is f from the Set {1, 2} to the Set {a, b}, while a Counterexample to a non-Bijective Function is the Identity Function.

References