Functions

Function f: A ➝ B — binary relation f of type A × B such that every x ∈ A relates to at most one y ∈ B

• domain: all possible input values
• domain of definition: all input values that actually produce defined output
• codomain: what could be the output of a function (like “integers” with f(x)=2x)
• range/image: what actually is the output of a function (like “even numbers” with f(x)=2x)

injective: if each element x of domain maps to at most one element y of codomain (“one-to-one”)

surjective: if each element x of domain maps to at least one element y of codomain; the range is the codomain (“onto”)

total: if the function is defined for all possible input values (domain)

bijective: if the function is total, injective, and surjective

Composition:

(g o f) of f: A ➝ B and g: B ➝ C

• $D_{g \circ f} \subseteq D_f$
• $R_{g \circ f} \subseteq R_g$

Inverse:

a function has an inverse only if it’s injective (one-to-one)