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