Definition:Dominate (Set Theory)

Set Theory
Let $S$ and $T$ be sets.

Then $S$ is dominated by set $T$ iff there exists an injection from $S$ to $T$.

This can be written: Sources differ.
 * $S \preccurlyeq T$
 * $S \le T$

If $S \preccurlyeq T$ then $T$ dominates $S$ and we can write $T \succcurlyeq S$.

Set $S$ is strictly dominated by set $T$ iff $S \preccurlyeq T$ but $T \not \preccurlyeq S$.

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

Number Sequences

 * Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.


 * Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.

Then $\left \langle {a_n} \right \rangle$ dominates $\left \langle {z_n} \right \rangle$ iff:
 * $\forall n \in \N: \left|{z_n}\right| \le a_n$