Math and science::Algebra::Aluffi

Identity, inclusion and restriction

This note describes 3 concepts: identity functions, inclusion functions and function restriction.

Identity function

Every set $A$ has a function whose graph is the subset of $A \times A$ consisting of the elements on the diagonal. This function is called the identity function on $A$ , denoted as ${id}_{A}$ .

{id}_{A} : A \to A, \forall a \in A, {id}_{A} (a) = a

The identity function can be generalized slightly to arrive at the inclusion function.

Inclusion function

Let $S$ be a subset of $A$ . The inclusion function $i : S \to A$ maps an element in $S$ to the same elements in $A$ .

i : S \to A, \forall s \in S, i (s) = s

For a given function $f$ , the inclusion function composes with $f$ to create a restriction.

Restriction

Let $f : A \to X$ be a function, and let $S \subseteq A$ be a subset of $A$ . The restriction of $f$ to $S$ , denoted as $f |_{S}$ is defined as:

f |_{S} : S \to X, \forall s \in S, f |_{S} = f (s)

The restriction $f |_{S}$ can be viewed as the composition $f \circ i$ , where $i : S \to A$ is the inclusion function.

Identity function and composition

The identity function is very special with respect to composition: for any function $f$ , both ${id}_{B} \circ f = f$ and $f \circ {id}_{A} = f$ . As a graphical representation, these two statements correspond to stating that the following two diagrams compute.

Source

Aluffi

18.09.2020