Injection implies Cardinal Inequality
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Theorem
Let $S$ and $T$ be sets.
Let $f: S \to T$ be an injection.
Let $\card T$ denote the cardinal number of $T$.
Let:
- $T \sim \card T$
where $\sim$ denotes set equivalence
Then:
- $\card S \le \card T$
Proof
Let $f \sqbrk S$ denote the image of $S$ under $f$.
\(\ds S\) | \(\sim\) | \(\ds f \sqbrk S\) | Set is Equivalent to Image under Injection | |||||||||||
\(\ds \) | \(\subseteq\) | \(\ds T\) | Image Preserves Subsets | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \card S\) | \(=\) | \(\ds \card {f \sqbrk S}\) | Equivalent Sets have Equal Cardinal Numbers | ||||||||||
\(\ds \) | \(\le\) | \(\ds \card T\) | Subset implies Cardinal Inequality |
$\blacksquare$