# Equivalence of Definitions of Dominate (Set Theory)

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## Theorem

Let $S, T$ be sets.

The following definitions of the concept of **Dominate** in the context of **Set Theory** are equivalent:

### 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 $T' \subseteq T$.

## Proof

### Definition 1 implies Definition 2

Let $f: S \to T$ be an injection.

By Injection to Image is Bijection, $f$ is a bijection from $S$ to the image of $f$.

$\Box$

### Definition 2 implies Definition 1

Let $T' \subseteq T$ such that there exists a bijection $f: S \to T'$.

Let $i: T' \to T$ be the inclusion of $T'$ in $T$.

Then by Composite of Injections is Injection, $i \circ f: S \to T$ is an injection.

$\blacksquare$

## Sources

- 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $2$. Set Theoretical Equivalence and Denumerability