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.