Subset

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Subset

In mathematics

Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, especially in set theory

Set theory

Set theory is the branch of mathematics that studies Set , which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics....
, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion.

i>A and B are sets and every element

Element (mathematics)

In mathematics, an element or member of a Set is any one of the distinct objects that make up that set....
of A is also an element of B, then:

or equivalently

If A is a subset of B, but A is not equal to B (i.e.

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In mathematics

Mathematics

Set theory

Definitions

If A and B are sets and every element

Element (mathematics)

In mathematics, an element or member of a Set is any one of the distinct objects that make up that set....
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.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

Power set

In mathematics, given a Set S, the power set of S, written , P, ℘ or Power set#Representing subsets as functions, is the set of all subsets of S....
of S).

The symbols ? and ?

Some authors use the symbols ? and ? to indicate "subset" and "superset" respectively, instead of the symbols ? and ?, but with the same meaning. So for example, for these authors, it is true of every set A that A ? A.

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 is a proper subset of .
Any set is a subset of itself, but not a proper subset.
The empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
, denoted by ∅, is also a subset of any given set X. (This statement is vacuously true
Vacuous truth

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 B" when in fact A is false....
.) The empty set is always a proper subset, except of itself.
The set is a proper subset of
The set of natural number
Natural number

In mathematics, a natural number can mean either an element of the Set = *n = = ? = ? ...
s is a proper subset of the set of rational number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s and the set of points in a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
is a proper subset of the set of points in a line
Line (mathematics)

In geometry, a line is a Curvature curve. When geometry is used to model the real world, lines are used to represent straight objects with negligible width and height....
. 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
Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
).

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. The ordinal number

Ordinal number

In set theory, an ordinal number, or just ordinal, is the order type of a well-order. They are usually identified with hereditarily transitive sets....
s 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].

For the power set

Power set

In mathematics, given a Set S, the power set of S, written , P, ℘ or Power set#Representing subsets as functions, is the set of all subsets of S....
2^S of a set S, the inclusion partial order is (up to an order isomorphism

Order isomorphism

In the mathematics field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets ....
) the Cartesian product

Cartesian product

In mathematics, the Cartesian product is a direct product of sets. The Cartesian product is named after Ren? Descartes, whose formulation of analytic geometry gave rise to this concept....
of k = |S| (the cardinality

Cardinality

In mathematics, the cardinality of a set is a measure of the "number of Element of the set". For example, the set A = contains 3 elements, and therefore A has a cardinality of 3....
of S) copies of the partial order on for which 0 < 1. This can be illustrated by enumerating S = and associating with each subset T ? S (which is to say with each element of 2^S) the k-tuple from ^k of which the ith coordinate is 1 if and only if s_i is a member of T.