Injective, Surjective & Bijective

Definition of Function

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 Function not Surjective
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 Function not Injective
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 Function Surjective Injective
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 Function
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.

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Set-Builder Notation

Function Transformations

Inverse Functions

Set Index