# Definition:Subset

## Definition

A set $S$ is a subset of a set $T$ if and only if all of the elements of $S$ are also elements of $T$, and it is written $S \subseteq T$.

If the elements of $S$ are not all also elements of $T$, then $S$ is not a subset of $T$:

$S \nsubseteq T$ means $\neg \left( {S \subseteq T}\right)$

For example, if $S = \left\{ {1, 2, 3} \right\}$ and $T = \left\{ {1, 2, 3, 4} \right\}$, then $S \subseteq T$.

So, if we can prove that if an element is in $S$ then it is also in $T$, then we have proved that $S$ is a subset of $T$.

That is:

$S \subseteq T \iff \forall x: \left({x \in S \implies x \in T}\right)$

In class-set theories in which sets are classes, a set $S$ is a subset of a class $T$ iff it is a subclass of $T$.

In class-set theories in which sets are not classes, writers may nevertheless say that a set $S$ is a subset of a class $T$ iff every element of $S$ is also an element of $T$.

### Superset

If $S$ is a subset of $T$, then $T$ is a superset of $S$.

This can be expressed by the notation $T \supseteq S$.

This can be interpreted as $T$ contains $S$.

Thus $S \subseteq T$ and $T \supseteq S$ mean the same thing.

## Also known as

When the concept 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 $S$

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

However, 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.

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 containing set, but this is rare and of doubtful value.

Notation in the literature can be confusing.

Many authors, for example 1968: A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis and 1971: Allan Clark: Elements of Abstract Algebra, use $\subset$.

If it is important with this usage to indicate that $S$ is a proper subset of $T$, the notation $S \subsetneq T$ or $T \supsetneq S$ can be used.

## Also see

• Results about subsets can be found here.

## Historical Note

The concept of the subset, set inclusion, was stated by Leibniz in his initial conception of symbolic logic.

## Notes

Note the difference between $x \in T$ and $S \subseteq T$.

We can see that is a subset of is a relation. Given any two sets $S$ and $T$, we can say that either $S$ is or is not a subset of $T$.