Functions on Sets

$f: A-> B, x -> f(x)$
$A =$ Domain
$B =$ Codomain

Image

For $C \subset A, f(C) := \{y \in B \mid \exists x \in C: f(x) = y \}$ is called the image of $C$ under $f$

Pre-image

For $D \subset B, f^{-1}(D) := \{x \in A | f(x) \in D \}$ is called the pre-image of $D$ under $f$

Mapping Properties

Let $f: A-> B$ be a function between non-empty sets. Then $f$ is called:

1. injective if $\forall x_1, x_2 \in A, x_1 \neq x_2$ (every y value has only one x value)
2. surjective if $\forall y \in B \exists x \in A: f(x)=y$ (all y values can be reaced by a certain x value)
3. bijective: both