Definition:Strictly Smaller Set

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Let $S$ and $T$ be sets.

$S$ is defined as being strictly smaller than $T$ if and only if:

$(1): \quad$ There exists a bijection from $S$ to a subset of $T$
$(2): \quad$ There exists no such bijection from $T$ to a subset of $S$.

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

$S < T$

Also known as

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

Also see