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:
 * $$\forall n \in \N: \left|{z_n}\right| \le a_n$$