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 $S$

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.


 * 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.

Notation in the literature can be confusing.

Many authors, for example and, 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

 * 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 $T$.