Subset

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.

"Superset" redirects here. For other uses, see Superset (disambiguation).

The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation.

A set A is a subset of B their intersection is equal to A.

if and only if

⊂ and ⊃ symbols[edit]

Some authors use the symbols $\subset$ and $\supset$ to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols $\subseteq$ and $\supseteq .$ ^[4] For example, for these authors, it is true of every set A that $A\subset A.$ (a reflexive relation).

Other authors prefer to use the symbols $\subset$ and $\supset$ to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols $\subsetneq$ and $\supsetneq .$ ^[5] This usage makes $\subseteq$ and $\subset$ analogous to the inequality symbols $\leq$ and $<.$ For example, if $x\leq y,$ then x may or may not equal y, but if $x<y,$ then x definitely does not equal y, and is less than y (an irreflexive relation). Similarly, using the convention that $\subset$ is proper subset, if $A\subseteq B,$ then A may or may not equal B, but if $A\subset B,$ then A definitely does not equal B.

The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions $A\subseteq B$ and $A\subsetneq B$ are true.

The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus $D\subseteq E$ is true, and $D\subsetneq E$ is not true (false).

Any set is a subset of itself, but not a proper subset. ( $X\subseteq X$ is true, and $X\subsetneq X$ is false for any set X.)

The set {x: x is a greater than 10} is a proper subset of {x: x is an odd number greater than 10}

prime number

The set of is a proper subset of the set of rational numbers; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.

natural numbers

The set of is a proper subset of the set of real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.

rational numbers

Another example in an Euler diagram:

– In geometry, set whose intersection with every line is a single line segment

Convex subset

– Partial order that arises as the subset-inclusion relation on some collection of objects

Inclusion order

– Connected open subset of a topological space

Region

– Decision problem in computer science

Subset sum problem

– System of elements that are subordinated to each other

Subsumptive containment

– Subset T of a topological vector space X where the linear span of T is a dense subset of X

Total subset

– Study of parts and the wholes they form

Mereology

(2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2.

Jech, Thomas

Media related to Subsets at Wikimedia Commons

"Subset". MathWorld.