Mappings and Functions¶

Let \(X,Y,Z\) denote sets.

Definition 21 (Mapping)

A mapping \(f:X \to Y\) is a rule associating every element in \(X\) with a unique member of \(Y\). \(X\) is the domain, and \(Y\) is the codomain of the mapping.

Definition 22 (Function Composition)

Given two mappings \(f:X \to Y\) and \(g:Y \to Z\), the composition, \(g \circ f: X \to Z\) is the mapping

\[(g \circ f)(x) = g(f(x)) \qquad \forall x \in X\]

Definition 23 (Identity Mapping)

The identity mapping

\[\idmap_X : X \to X\]

which is defined by

\[\idmap_X(x) = x \quad \forall x \in X\]

Definition 24 (Zero Mapping)

Provided \(Y\) contains a zero element \(0\), the zero mapping, \(0_x:X \to Y\) is defined

\[0_x(x) = 0 \quad \forall x \in X\]

Definition 25 (Injective, surjective, and bijective mappings)

A mapping \(f:X \to Y\) is injective if for all \(x_1, x_2 \in X\),

\[f(x_1) = f(x_2) \implies x_1 = x_2\]

A mapping \(f: X \to Y\) is called surjective if, for every \(y \in Y\) there exists at least one \(x \in X\) for which \(y = f(x)\).

A mapping is bijective if it is both injective and surjective.

Observation 1 (Inverse mappings)

A mapping \(f:X \to Y\) is bijective iff there is an inverse mapping

\[h: Y \to X\]

such that

\[h \circ f = \idmap_X \quad \text{and} \quad f \circ h = \idmap_Y\]