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

• Equivalence of Definitions of Smaller Set

• Definition:Strictly Smaller Set

• Definition:Larger Set
• Definition:Strictly Larger Set

• Results about smaller set can be found here.