Definition:Subset

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Definition

Let $S$ and $T$ be sets.

$S$ is a subset of a set $T$ 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 $\neg \paren {S \subseteq T}$



In class-set theories in which sets are classes, a set $S$ is a subset of a class $T$ if and only if 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 $T$ if and only if 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 $T$ includes $S$, or (more rarely) $T$ 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.

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.


Notation in the literature can be confusing.

Many authors, for example A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis and 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.


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 $\Z$:

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


Also see


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 $T$.


  • 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