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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.