Definition:Subset

Definition
Let $S$ and $T$ be sets.

$S$ is a subset of a set $T$ all of the elements of $S$ are also elements of $T$.

This is denoted:
 * $S \subseteq T$

That is:
 * $S \subseteq T \iff \forall x: \paren {x \in S \implies x \in T}$

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


 * $S \nsubseteq T$ means $\neg \paren {S \subseteq T}$

Euler Diagram
In class-set theories in which sets are classes, a set $S$ is a subset of a class $T$ 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 $T$ every element of $S$ is also an element of $T$.

Also known as
When the concept was first raised by, 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.

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.

Also see

 * Definition:Proper Subset

Note the difference between:
 * an element: $x \in T$

and:
 * a subset: $S \subseteq T$.

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 $T$.