# Definition:Subset

## Contents

## 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

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