# Definition:Smaller Set

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## 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.