# Definition:Dominate (Set Theory)

## 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:

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