# Definition:Dominate (Set Theory)

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## Definition

Let $S$ and $T$ be sets.

### Definition 1

Then **$S$ is dominated by $T$** if and only if there exists an injection from $S$ to $T$.

### Definition 2

Then **$S$ is dominated by $T$** if and only if $S$ is equivalent to some subset of $T$.

That is, if and only if there exists a bijection $f: S \to T'$ for some $T' \subseteq T$.

### Strictly Dominated

**$S$ is strictly dominated by set $T$** if and only if $S \preccurlyeq T$ but $\neg T \preccurlyeq S$.

This can be written $S \prec T$ or $S < T$.

## Also denoted as

Alternatives for $S \preccurlyeq T$ include:

- $S \le T$
- $S \leqslant T$