Definition:Dominate
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Definition
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:
- $S \preccurlyeq T$
- $S \le T$
Sources differ.
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:
- $\forall n \in \N: \left|{z_n}\right| \le a_n$
Split these definitions out into two separate disambiguated pages, as they mean completely different things.
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