Inclusion Mapping is Restriction of Identity
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Theorem
Let $T$ be a set.
Let $S \subseteq T$ be a subset of $T$.
Let $i_S: S \to T$ be the inclusion mapping on $S$.
Then $i_S$ is the restriction of the identity mapping $I_T: T \to T$ on $T$.
Proof
By definition of inclusion mapping:
- $i_S: S \to T: \forall x \in S: \map {i_S} x = x$
By definition of identity mapping:
- $I_T: T \to T: \forall x \in T: \map {I_T} x = x$
The result follows by definition of restriction of mapping.
$\blacksquare$
Sources
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.3$: Mappings