# Definition:Smaller Set

## Definition

Let $S$ and $T$ be sets.

### Definition 1

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

### Definition 2

$S$ is defined as being smaller than $T$ if and only if there exists an injection from $S$ into $T$.

### Notation

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

$S \le T$

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

• Results about smaller set can be found here.