# Subset

supersetproper subsetsubsetsinclusionset inclusionpropercontainmentproper 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 ⊆ 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 ⊆ 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), 𝒫(S), ℘(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 ⊆ 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 × B. Since the latter set is ordered by inclusion, each relation has a place in the lattice of subsets of A × 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 "P→Q" all mean that P is a subset, either proper or improper, of Q. "P if Q", "if Q then P", and Q→P 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 ⊆ and ⊂ analogous to the inequality symbols ≤ 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 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].
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 exists

### Prime number

primeprime factorprime numbers

### Rational number

rationalrational numbersrationals

### Line (geometry)

linestraight linelines