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In , especially in , a 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 on sets.

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


Definitions
If A and B are sets and every of A is also an element of B, then:
* A is a subset of (or is included in) B, denoted by A \subseteq B,
or equivalently
* B is a superset of (or includes) A, denoted by B \supseteq A.

If A is a subset of B, but A is not to B (i.e. 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\subsetneq B.
or equivalently
* B is a proper superset of A; this is written as B\supsetneq A.

For any set S, the inclusion ⊆ is a on the set \mathcal{P}(S) of all subsets of S (the 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 the 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.


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 { }, 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 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 is a proper subset of the set of points in a . These are examples in which both the part and the whole are infinite, and the part has the same (number of elements) as the whole; such cases can tax one's intuition.


Other properties of inclusion
Inclusion is the canonical in the sense that every partially ordered set ( X, \preceq) is to some collection of sets ordered by inclusion. The are a simple example—if each ordinal n is identified with the set n of all ordinals less than or equal to n, then ab if and only if ab.

For the \mathcal{P}(S) of a set S, the inclusion partial order is (up to an ) the of k = | S| (the 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 TS (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.


See also


External links

References
    ^ (2017). 9783540440857, Springer-Verlag.

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