Subset

 ( Basic Concepts In Set Theory ) 

In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment.

The subset relation defines a partial order on sets.

The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion.

* A is a subset of (or is included in) B, denoted by $A\; \backslash subseteq\; B$,

or equivalently

* B is a superset of (or includes) A, denoted by $B\; \backslash supseteq\; A.$

If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then

* A is also a proper (or strict) subset of B; this is written as $A\backslash subsetneq\; B.$

or equivalently

* B is a proper superset of A; this is written as $B\backslash supsetneq\; A.$

For any set S, the inclusion relation ⊆ is a partial order on the set $\backslash mathcal\{P\}(S)$ of all subsets of S (the power set of S).

Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, in place of ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤  y then x may be equal to y, or maybe not, but if x <  y, then x definitely does not equal y, and is strictly less than y. Similarly, using the "⊂ means proper subset" convention, if A ⊆  B, then A may or may not be equal to B, but if A ⊂  B, then A is definitely not equal to B.

• The set {1, 2} is a proper subset of {1, 2, 3}.
• Any set is a subset of itself, but not a proper subset.
• The empty set { }, denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any set except itself.
• The set { x: x is a prime number greater than 10} is a proper subset of { x: x is an odd number greater than 10}
• The set of is a proper subset of the set of , and the set of points in a line segment is a proper subset of the set of points in a line. These are examples in which both the part and the whole are infinite, and the part has the same cardinality (number of elements) as the whole; such cases can tax one's intuition.

For the power set $\backslash mathcal\{P\}(S)$ of a set S, the inclusion partial order is (up to an order isomorphism) the Cartesian product of k = | S| (the cardinality of S) copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating S = { s₁, s₂, …, s _k} and associating with each subset T ⊆ S (which is to say with each element of 2 ^S) the k-tuple from {0,1} ^k of which the ith coordinate is 1 if and only if s _i is a member of T.

• Containment order

• ξ1

^ Jech, Thomas (2021). 9783540440857, Springer-Verlag. ISBN 9783540440857