Injective, surjective, bijective functions

Injective functions

• Also know as one-to-one
  • This is if every input to the function has only 1, unique output
• Let , is called a injective function if
  • for each , , at most one solution
  • , then

Example: let , as

Surjective functions

• This can be described as ‘onto’
  • Basically, if every element in the codomain has a solution
• Let . is called surjective if
  • for each , the equation has at least one solution

Example: , (inc.)

Bijective functions

• Also known as an invertible function
• Let . is called bijective if
  • is one-one
  • for each , has exactly one solution

Example: . , the identity

• Key point: to prove that a function is bijective, you have to prove that it is both injective and surjective