Identity Mapping is Injection
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Theorem
On any set $S$, the identity mapping $I_S: S \to S$ is an injection.
Proof
From the definition of the identity mapping:
- $\forall x \in S: \map {I_S} x = x$
So:
- $\map {I_S} x = \map {I_S} y \implies x = y$
So by definition $I_S$ is an injection.
$\blacksquare$
Also see
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Composition of Functions
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Example $5.3$