Identity Mapping is Injection

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 $5.3$
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