▗▄▄▄▖▗▞▀▚▖▄▄▄▄  ▄    ■  ▗▞▀▜▌▄   ▄  ▄▄▄       ▗▄▖    ▗▖ ▄▄▄  
  █  ▐▛▀▀▘█ █ █ ▄ ▗▄▟▙▄▖▝▚▄▟▌█   █ █   █     ▐▌ ▐▌   ▗▖█   █ 
  █  ▝▚▄▄▖█   █ █   ▐▌        ▀▀▀█ ▀▄▄▄▀     ▐▌ ▐▌▄  ▐▌▀▄▄▄▀ 
  █             █   ▐▌       ▄   █           ▝▚▄▞▘▀▄▄▞▘      
                    ▐▌        ▀▀▀                            
                                                             

# Functions

Here are some useful definitions regarding functions. Consider a function from a set `A` to a set `B` which we will denote `f: A to B`.

• We call `A` the domain of `f`.
• We call `B` the codomain of `f`.
• For a subset `X sub A` we call the set `f(X) := {b in B | exists a in A` s.t `b = f(a)}` the image of `X` under `f`.
• We call the set `f(A)` the range or image of `f`.
• For a subset `Y subseteq B` we call the set `f^-1 (Y):= {a in A | f(a) in Y}` the preimage or inverse image of `Y` under `f`.
• We say f in injective (is an injection) if `forall x_1,x_2 in A` we have `x_1 ne x_2 implies f(x_1) ne f(x_2).`
• We say `f` is surjective (is a surjection) if `forall y in B` we have `exists x in A` s.t. `f(x) = y`.
• We say `f` is bijective (is a bijection) if it is both injective and surjective.