Subset
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.
A k-subset is a subset with k elements.
One can prove the statement
by applying a proof technique known as the element argument[2]:Let sets A and B be given.
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
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: The empty set, written
has no elements, and therefore is vacuously a subset of any set X.
Some authors use the symbols
to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols
[4] For example, for these authors, it is true of every set A that
(a reflexive relation).
Other authors prefer to use the symbols
to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols
analogous to the inequality symbols
then x definitely does not equal y, and is less than y (an irreflexive relation).
Another example in an Euler diagram: The set of all subsets of
is called its power set, and is denoted by
is a partial order on the set
We may also partially order
by reverse set inclusion by defining
of a set S, the inclusion partial order is—up to an order isomorphism—the Cartesian product of
(the cardinality of S) copies of the partial order on
This can be illustrated by enumerating
, in analogue with the notation for binomial coefficients, which count the number of
In set theory, the notation
is a transfinite cardinal number.
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ).
