Identity Mapping is Idempotent
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Theorem
Let $S$ be a set.
Let $I_S: S \to S$ be the identity mapping on $S$.
Then $I_S$ is idempotent:
- $I_S \circ I_S = I_S$
Proof
From Identity Mapping is Left Identity:
- $I_S \circ f = f$
for all mappings $f: S \to S$.
From Identity Mapping is Right Identity:
- $f \circ I_S = f$
for all mappings $f: S \to S$.
Substituting $I_S$ for $f$ in either one and the result follows.
$\blacksquare$
Sources
- 1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.14$: Composition of Functions: Theorem $14.7$