# Definition:Subset

## Contents

## 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*

- -- 1975: T.S. Blyth:

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

- 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 - 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): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 2$: Sets and Subsets - 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 - 1986: Geoffrey Grimmett and Dominic Welsh:
*Probability: An Introduction*... (previous) ... (next): $\S 1.2$: Outcomes and events - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*: Entry:**subset**or**subclass** - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set? - 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 $\text{A}.3$