# Definition:Smaller Set

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## Definition

Let $S$ and $T$ be sets.

$S$ is defined as being **smaller than** $T$ if and only if there exists a bijection from $S$ to a subset of $T$.

**$S$ is smaller than $T$** can be denoted:

- $S \le T$

## Also defined as

Some sources define $S \le T$ if and only if there exists an injection from $S$ into $T$.

## Also denoted as

Some sources denote this using an explicit ordering on the cardinalities of the sets in question:

- $\card S \le \card T$

## Also known as

If $S$ is **smaller than** $T$, then $S$ is said to be of **lower cardinality** or **smaller cardinality** than $T$.

## Also see

## Sources

- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $3$: Cardinality: Definition $2$ - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 4$ Larger and smaller