Definition:Subset

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Definition

Let $S$ and $T$ be sets.

$S$ is a subset of $T$ if and only if 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

The statement that $T$ is a subset of $S$ can be illustrated in the following Euler diagram.

EulerDiagram2Subset.png


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 $T$ includes $S$, or (more rarely) $T$ contains $S$.

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


Subset of Class

A set $S$ is a subset of a class $T$ if and only if it is a subclass of $T$.

Alternatively, a set $S$ is a subset of a class $T$ if and only if every element of $S$ is also an element of $T$.


Subset Relation

The subset relation, written $\subseteq$, is the class of all ordered pairs $\tuple {x, y}$ such that $x$ is a subset of $y$.


Notation

Notation in the literature for the concept of a subset can be confusing.

On $\mathsf{Pr} \infty \mathsf{fWiki}$, the convention is as follows:

$\subseteq$ is used for the general subset
$\subsetneq$ or $\subsetneqq$ is used for the concept of a proper subset.

The notation $\subset$ is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$, on the grounds that it can mean either.


Examples

British People are Subset of People

Let $B$ denote the set of British people.

Let $P$ denote the set of people.

Then $B$ is a proper subset of $P$:

$B \subsetneq P$


Subset of Alphabet

Let $S$ denote the capital letters of the (English) alphabet:

$S = \set {A, B, C, D, \dotsc, Z}$


Then $\set {A, B, C}$ is a subset of $S$:

$\set {A, B, C} \subseteq S$


Integers are Subset of Real Numbers

The set of integers $\Z$ is a proper subset of the set of real numbers $\R$:

$\Z \subsetneq \R$


Initial Segment is Subset of Integers

The (one-based) initial segment of the natural numbers $\N^*_{\le n}$ of integers $\Z$ is a proper subset of the set of integers $\Z$:

$\N^*_{\le n} \subsetneq \Z$


Even Numbers form Subset of Integers

The set of even integers forms a subset of the set of integers $\Z$.


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.


Some sources use the term subclass to mean the same as subset, but on $\mathsf{Pr} \infty \mathsf{fWiki}$ we insist that a class is NOT the same thing as a set.

Burn this usage with fire.


Also see


Note the difference between:

an element: $x \in T$

and:

a subset: $S \subseteq T$.


  • 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.


Sources

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