# Injective, Surjective & Bijective

## Definition of Function

A General Function

• Points each member of “A” to a member of “B”.
• A member of “A” only points one member of “B”. (one-to-many is not allowed. Example: f(x) = 3 or 4 is not allowed)
• 2 or more members of “A” can point to the same “B” (many-to-one is allowed. Example: f(4) = f(5) = 4 is allowed)

## Injective

Injective means

• No 2 or more members of “A” point to the same “B”.
• Many-to-one is NOT allowed (Example: f(3) = f(4) = 2 is NOT allowed).
• One-to-many is NOT allowed too (Because it’s a function)
• Member(s) of “B” without a matching “A” is allowed
• Injective is also called “One-to-One

A function f is injective if and only if whenever f(x) = f(y), x = y.

## Surjective

Surjective means

• Every member of “B” has at least 1 matching “A” (can has more than 1).
• Member(s) of “B” without a matching “A” is NOT allowed

f is surjective if and only if f(A) = B

A function f (from set A to B) is surjective if and only if for every y in B, there is at least one x in A such that f(x) = y

## Bijective

Bijective means

• Both Injective and Surjective together.
• A perfect “one-to-one correspondence” between the members of the sets. (Don’t get that confused with “One-to-One” used in injective).

A function f (from set A to B) is bijective if, for every y in B, there is exactly one x in A such that f(x) = y

## Inverse

Bijective means

Bijection function is also known as invertible function because it has inverse function property.

It is a function which assigns to b, a unique element a such that f(a) = b. hence f-1 (b) = a.