Definition:Subset

Definition

A set $S$ is a subset of a set $T$ iff 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 $\neg \left( {S \subseteq T}\right)$

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

$S \subseteq T \iff \forall x: \left({ x \in S \implies x \in T}\right)$

Superset

If $S$ is a subset of $T$, then that means $T$ is a superset of $S$, which can be expressed by the notation $T \supseteq S$. This can be interpreted as $T$ contains $S$.

Thus $S \subseteq T$ and $T \supseteq S$ mean the same thing.

Alternative names

$S \subseteq T$ 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 operator, or (more rarely) the containment operator.

Also see

• Compare the concept of a proper subset.

Notation in the literature can be confusing. Many authors, for example A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968) and Allan Clark: Elements of Abstract Algebra (1971), 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.

• Results about subsets can be found here.

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

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 1$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 1$: The Axiom of Extension
• W.E. Deskins: Abstract Algebra (1964): $\S 1.1$: Definition $1.1$
• Steven A. Gaal: Point Set Topology (1964)... (previous)... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.2$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.1$
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 3$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.1$
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.1$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 6$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.1$