Definition:Subset/Also known as

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Subset: Also known as

When the concept of the subset was first raised by Georg Cantor, he used the terms part and partial aggregate for this concept.


$S \subseteq T$ can also be read as:

$S$ is contained in $T$, or $T$ contains $S$
$S$ is included in $T$, or $T$ includes $S$

The term weakly includes or weakly contains can sometimes be seen here, to distinguish it from strict inclusion.

Hence $\subseteq$ is also called the inclusion relation, or (more rarely) the containment relation.

The term weakly includes or weakly contains can sometimes be seen here, to distinguish it from strict inclusion.


Beware of this usage: $T$ contains $S$ can also be interpreted as $S$ is an element of $T$.

Such is the scope for misinterpretation that it is mandatory that further explanation is added to make it clear whether you mean subset or element.

A common way to do so is to append "as a subset" to the phrase.

We also describe this situation by saying that $E$ is included in $F$ or that $E$ is contained in $F$, though the latter terminology is better avoided.
-- 1975: T.S. Blyth: Set Theory and Abstract Algebra


In contrast with the concept of a proper subset, the term improper subset can occasionally be seen to mean a subset which may equal its superset, but this is rare and of doubtful value.


Sources

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