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 [...] consisting of [what elements?]. This function is called the identity function on $$A$$, denoted as $$\operatorname{id_A}$$.

$\operatorname{id_A} : A \to A, \quad \forall a \in A, \, \operatorname{id_A}(a) = a$

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 $$A$$.

[$i : S \to A, \quad \forall s \in S, \, ? = \, \, ?$]

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:

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

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