- Subset
In

mathematics , especially inset theory , 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**.**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 $A\; subseteq\; B$,: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 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 $Asubsetneq\; B.$:or equivalently:* "B" is a proper superset of "A"; this is written as $Bsupsetneq\; A.$For any set "S", the inclusion relation ⊆ is a

partial order on the set 2^{"S"}of all subsets of "S" (thepower set 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 $subsetneq$ and $supsetneq.$ 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.

* Theempty set , written Unicode|∅, is also a subset of any given set "X". (This statement is vacuously true.) The empty set is always a proper subset, except of itself.

* The set {"x": "x" is aprime number greater than 2000} is a proper subset of {"x": "x" is an odd number greater than 1000}

* The set ofnatural number s is a proper subset of the set ofrational number s and the set of points in aline segment is a proper subset of the set of points in a line. 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).**Other properties of inclusion**Inclusion is the canonical

partial order in the sense that every partially ordered set ("X", $preceq$) isisomorphic to some collection of sets ordered by inclusion. Theordinal number 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 2^{"S"}of a set "S", the inclusion partial order is (up to anorder isomorphism ) theCartesian product of "k" = |"S"| (thecardinality of "S") copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating "S" = {"s"_{1}, "s"_{2}, …, "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 "i"th coordinate is 1 if and only if "s"_{"i"}is a member of "T".**References***

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