# Subset

**supersetproper subsetsubsetsinclusionset inclusionproperâcontainmentproper supersetsubset inclusion**

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B.wikipedia

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### Partially ordered set

**partial orderposetpartially ordered**

The subset relation defines a partial order on sets. For any set S, the inclusion relation â is a partial order on the set of all subsets of S (the power set of S) defined by.

### Element (mathematics)

**elementelementsset membership**

That is, all elements of A are also elements of B.

Sets of elements of A, for example \{1, 2\}, are subsets of A.

### Power set

**powerset2all subsets**

For any set S, the inclusion relation â is a partial order on the set of all subsets of S (the power set of S) defined by. For the power set 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, the power set (or powerset) of any set S is the set of all subsets of S, including the empty set and S itself, variously denoted as (S), đ«(S), â(S) (using the "Weierstrass p"),

### Mathematics

**mathematicalmathmathematician**

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B.

As the number system is further developed, the integers are recognized as a subset of the rational numbers ("fractions").

### Binary relation

**relationrelationsidentity relation**

For any set S, the inclusion relation â is a partial order on the set of all subsets of S (the power set of S) defined by.

A binary relation over A and B is an element of the power set of A Ă B. Since the latter set is ordered by inclusion, each relation has a place in the lattice of subsets of A Ă B.

### Euler diagram

**EulerEuler diagramsEuler's diagrams**

Another example in an Euler diagram:

A curve that is contained completely within the interior of another represents a subset of it.

### Empty set

**emptynon-emptynonempty**

By the definition of subset, the empty set is a subset of any set A.

### Line segment

**segmentline segmentssegments**

If V is a vector space over \mathbb{R} or \mathbb{C}, and L is a subset of V, then L is a line segment if L can be parameterized as

### Cardinality

**cardinalitiesnumber of elementssize**

For the power set 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.

These results are highly counterintuitive, because they imply that there exist proper subsets and proper supersets of an infinite set S that have the same size as S, although S contains elements that do not belong to its subsets, and the supersets of S contain elements that are not included in it.

### If and only if

**iffif, and only ifmaterial equivalence**

"P only if Q", "if P then Q", and "PâQ" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and QâP all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.

### Inequality (mathematics)

**inequalityinequalitiesLess than**

This usage makes â and â analogous to the inequality symbols â€ and

### Containment order

**containment**

* Containment order

In the mathematical field of order theory, a containment order is the partial order that arises as the subset-containment relation on some collection of objects.

### Ordinal number

**ordinalordinalsordinal numbers**

The ordinal numbers are a simple exampleâif each ordinal n is identified with the set [n] of all ordinals less than or equal to n, then a â€ b if and only if [a] â [b].

Given two ordinals S and T, S is an element of T if and only if S is a proper subset of T.

### Cartesian product

**productCartesian squareCartesian power**

For the power set 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.

Other properties related with subsets are:

### Real number

**realrealsreal-valued**

The well-ordering theorem implies that the real numbers can be well-ordered if the axiom of choice is assumed: there exists a total order on R with the property that every non-empty subset of R has a least element in this ordering.

### Natural number

**natural numberspositive integerpositive integers**

is a subset of

### Set (mathematics)

**setsetsmathematical set**

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained in B.

### Boolean algebra (structure)

**Boolean algebraBoolean algebrasBoolean lattice**

In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the meet and join are given by intersection and union.

### Intersection (set theory)

**intersectionintersectionsset intersection**

In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the meet and join are given by intersection and union.

### Union (set theory)

**unionset unionunions**

In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the meet and join are given by intersection and union.

### Equality (mathematics)

**equalityequalequalities**

### Existential quantification

**existential quantifierthere existsâ**

### Prime number

**primeprime factorprime numbers**

### Rational number

**rationalrational numbersrationals**

### Line (geometry)

**linestraight linelines**