Euler diagram showing
A is a subset of B

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Syllogism-Set-Diagramsjpg|thumb|Examples of small Venn diagrams with shaded regions representing Empty sets that are easily transformed into Euler diagrams Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion.

Definitions

If A and B are sets and every element of A is also an element of B, then:

• A is a subset of (or is included in) B, denoted by ,

or equivalently

• B is a superset of (or includes) A, denoted by

If A is a subset of B, but A is not equal to B (i. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the Equality is the paradigmatic example of the more general concept of Equivalence relations on a set those binary relations which are reflexive, symmetric e. there exists at least one element of B not contained in A), then

• A is also a proper (or strict) subset of B; this is written as

or equivalently

• B is a proper superset of A; this is written as

For any set S, the inclusion relation ⊆ is a partial order on the set 2^S of all subsets of S (the power set of S). In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S)

The symbols ⊂ and ⊃

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 ≤ 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, but 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.

Examples

• 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, written ∅, is also a subset of any given set X. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members (This statement is vacuously true, see proof below) The empty set is always a proper subset, except of itself. A vacuous truth is a truth that is devoid of content because it asserts something about all members of a class that is empty or because it says "If  A  then 
• The set {x : x is a prime number greater than 2000} is a proper subset of {x : x is an odd number greater than 1000}
• The set of natural numbers is a proper subset of the set of rational numbers and the set of points in a line segment is a proper subset of the set of points in a line. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points These are counter-intuitive examples in which both the part and the whole are infinite, and the part has the same number of elements as the whole (see Cardinality of infinite sets). In Mathematics, the cardinality of a set is a measure of the "number of elements of the set"

Other properties of inclusion

Inclusion is the canonical partial order in the sense that every partially ordered set (X, ) is isomorphic to some collection of sets ordered by inclusion. In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective The ordinal numbers are a simple example—if each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then a ≤ b if and only if [a] ⊆ [b]. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.

For the power set 2^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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In the mathematical field of Order theory an order isomorphism is a special kind of Monotone function that constitutes a suitable notion of Isomorphism Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" 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.