Classifying functions

A function can have multiple properties, which describe the general behavior under certain conditions.

Injective

The function is reversable.

Definition: A function f: A \to B is injective, if \forall x_1, x_2 \in A, x_1 \neq x_2 : f(x_1) \neq f(x_2)

Note: This is a one way relation; not every B can be mapped to an A.

There are no collisions when the function is injectiv.

If a function f: A \to B is injective; |A| \leq |B|

Surjective

Definition: A function f:A \to B is surjective, if \forall b \in B there exists at least one a \in A such that f(a) = b

If a function f: A \to B is bijective; |A| \geq |B|

Bijective

Definition: A function f: A \to B is called bijectiv if it is injective and surjective, meaning, \forall b \in B there exist exactly one a \in A such that f(a) = b.

If a function f: A \to B is bijectiv; |A| = |B|

Every encryption method E has an inverse function which can decrypt C using K, meaning, every encryption method is a bijectiv function.