# 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 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

- 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

- 1915: Georg Cantor:
*Contributions to the Founding of the Theory of Transfinite Numbers*... (previous) ... (next): First Article: $\S 1$: The Conception of Power or Cardinal Number - 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 1$: Operations on Sets - 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection - 1960: Paul R. Halmos:
*Naive Set Theory*... (previous) ... (next): $\S 1$: The Axiom of Extension - 1962: Bert Mendelson:
*Introduction to Topology*... (previous) ... (next): $\S 1.2$: Sets and Subsets - 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.1$: Definition $1.1$ - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.2$. Subsets - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory - 1966: Richard A. Dean:
*Elements of Abstract Algebra*... (previous) ... (next): $\S 0.2$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Introduction: Set-Theoretic Notation - 1968: A.N. Kolmogorov and S.V. Fomin:
*Introductory Real Analysis*... (previous) ... (next): $\S 1.1$: Basic definitions - 1968: Ian D. Macdonald:
*The Theory of Groups*... (previous) ... (next): Appendix: Elementary set and number theory - 1970: Avner Friedman:
*Foundations of Modern Analysis*... (previous) ... (next): $\S 1.1$: Rings and Algebras - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: The Notation and Terminology of Set Theory: $\S 3$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.3$: Subsets: Definition $3.1$ - 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 5.9$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Sets - 1974: Murray R. Spiegel:
*Theory and Problems of Advanced Calculus*(SI ed.) ... (previous) ... (next): Chapter $1$: Numbers: Sets - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 1$ - 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): Notation and Terminology - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1$: Real Numbers: $\S 1.1$: Set Notation - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.1$: Sets and Subsets - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 6$: Subsets - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Point Sets: $14.$ - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets - 1983: George F. Simmons:
*Introduction to Topology and Modern Analysis*... (previous) ... (next): $\S 1$: Sets and Set Inclusion - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*: Entry:**Subset** - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1.1$: What is a Set? - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.2$: Outcomes and events - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (previous) ... (next): Appendix $\text{A}.1$: Sets - 1999: András Hajnal and Peter Hamburger:
*Set Theory*... (previous) ... (next): $1$. Notation, Conventions: $5$: Definition $1.1$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): Appendix $\text{A}$: Set Theory: Sets and Subsets - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.2$: Elements, my dear Watson - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): Appendix $\text{A}.2$: Definition $\text{A}.3$