Relations & Functions in Mathematics

What is a Function?

A function is a special relation between two sets where each element in the domain is related to exactly one element in the co-domain.
If f: A ---> B, then for every a in A, there exists one and only one b in B such that f(a) = b.

Types of Functions:

1. One-One Function (Injective Function)

A function f: A ---> B is called a One-One function if different elements in set A map to different elements in set B.
In other words, if f(a1) = f(a2) implies a1 = a2, then the function is One-One.

Example (One-One):
Let f(x) = 2x, and A = {1, 2, 3}
Then:
f(1) = 2
f(2) = 4
f(3) = 6

All outputs are different. So, this function is One-One.

Example (Not One-One):
Let f(x) = x², and A = {-2, 2}
Then:
f(-2) = 4
f(2) = 4

Same output for different inputs. So, the function is not One-One.

2. Onto Function (Surjective Function)

A function f: A ---> B is called an Onto function if every element of set B has at least one pre-image in set A.
In other words, the range of the function is equal to the co-domain.

Example (Onto):
Let f(x) = x + 1
A = {1, 2, 3}, B = {2, 3, 4}
Then:
f(1) = 2
f(2) = 3
f(3) = 4

All elements of B are covered. So, the function is Onto.

Example (Not Onto):
Let f(x) = x²
A = {-2, -1, 0, 1, 2}, B = {-1, 0, 1, 4}

Output of f(x) is always non-negative, so -1 is not covered.
So, this function is not Onto.

3. Bijective Function

A function is called Bijective if it is both One-One and Onto. That means: 

• Every element of A maps to a unique element of B (One-One)
• Every element of B is covered (Onto)