Mapping from Singleton is Injection
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Theorem
Let $f: S \to T$ be a mapping.
Let $S$ be a singleton.
Then $f$ is an injection.
Proof
Let $S = \set s$.
For $f$ to be an injection, all we need to do is show:
- $\forall x_1, x_2 \in S: \map f {x_1} = \map f {x_2} \implies x_1 = x_2$
But as $S$ is a singleton, it follows that:
- $x_1 = x_2 = s$
Hence the result.
$\blacksquare$