Definition:Subset

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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 $\neg \paren {S \subseteq T}$


For example, if $S = \set {1, 2, 3}$ and $T = \set {1, 2, 3, 4}$, 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:

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


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 $T$ 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.


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.


Examples

Integers are Subset of Real Numbers

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

$\Z \subsetneq \R$


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 $P$:

$B \subsetneq P$


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

  • 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 $T$.


Sources