
In
mathematics, a
set ''A'' is a subset of a set ''B'' if all
elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''.
The subset relation defines a
partial order on sets. In fact, the subsets of a given set form a
Boolean algebra under the subset relation, in which the
join and meet are given by
intersection and
union, and the subset relation itself is the
Boolean inclusion relation.
Definitions
If ''A'' and ''B'' are sets and every
element of ''A'' is also an element of ''B'', then:
:*''A'' is a subset of ''B'', denoted by
or equivalently
:* ''B'' is a superset of ''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 which is not an element of ''A''), then:
:*''A'' is a proper (or strict) subset of ''B'', denoted by
(or
). Or equivalently,
:* ''B'' is a proper (or strict) superset of ''A'', denoted by
(or
).
:* The
empty set, written or ∅, is a subset of any set ''X'' and a proper subset of any set except itself.
For any set ''S'', the inclusion
relation ⊆ is a
partial order on the set
(the
power set of ''S''—the set of all subsets of ''S'') defined by
. We may also partially order
by reverse set inclusion by defining
When quantified, is represented as .
We can prove the statement by applying a proof technique known as the element argument:
Let sets ''A'' and ''B'' be given. To prove that ,
# suppose that ''a'' is a particular but arbitrarily chosen element of ''A,''
# show that ''a'' is an element of ''B''.
The validity of this technique can be seen as a consequence of
Universal generalization: the technique shows for an arbitrarily chosen element ''c''. Universal generalisation then implies , which is equivalent to , as stated above.
Properties
* A set ''A'' is a subset of ''B''
if and only if their intersection is equal to A.
:Formally:
:
* A set ''A'' is a subset of ''B'' if and only if their union is equal to B.
:Formally:
:
* A finite set ''A'' is a subset of ''B'', if and only if the
cardinality of their intersection is equal to the cardinality of A.
:Formally:
:
⊂ and ⊃ symbols
Some authors use the symbols ⊂ and ⊃ to indicate ''subset'' and ''superset'' respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇. For example, for these authors, it is true of every set ''A'' that .
Other authors prefer to use the symbols ⊂ and ⊃ to indicate ''proper'' (also called strict) subset and ''proper'' superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋.
This usage makes ⊆ and ⊂ analogous to the
inequality symbols ≤ and <. For example, if , then ''x'' may or may not equal ''y'', but if , then ''x'' definitely does not equal ''y'', and ''is'' less than ''y''. Similarly, using the convention that ⊂ is proper subset, if , then ''A'' may or may not equal ''B'', but if , then ''A'' definitely does not equal ''B''.
Examples of subsets

* The set A = is a proper subset of B = , thus both expressions A ⊆ B and A ⊊ B are true.
* The set D = is a subset (but ''not'' a proper subset) of E = , thus D ⊆ E is true, and D ⊊ E is not true (false).
* Any set is a subset of itself, but not a proper subset. (X ⊆ X is true, and X ⊊ X is false for any set X.)
* The set is a proper subset of
* The set of
natural numbers is a proper subset of the set of
rational numbers; likewise, the set of points in a
line segment is a proper subset of the set of points in a
line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same
cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition.
* The set of
rational numbers is a proper subset of the set of
real numbers. In this example, both sets are infinite, but the latter set has a larger cardinality (or ''power'') than the former set.
Another example in an
Euler diagram:
File:Example of A is a proper subset of B.svg|A is a proper subset of B
File:Example of C is no proper subset of B.svg|C is a subset but not a proper subset of B
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 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''
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 for which 0 < 1. This can be illustrated by enumerating ''S'' = , and associating with each subset ''T'' ⊆ ''S'' (i.e., each element of 2
''S'') the ''k''-tuple from
''k'', of which the ''i''th coordinate is 1 if and only if ''s''
''i'' is a member of ''T''.
See also
*
Inclusion order
*
Region
*
Subset sum problem
*
Subsumptive containment
*
Total subset
References
Bibliography
*
External links
*
*
{{Common logical symbols
Category:Basic concepts in set theory