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

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).

Similarly, using the convention that

is proper subset, if

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

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

, and associating with each subset

, in analogue with the notation for binomial coefficients, which count the number of

In set theory, the notation

is a transfinite cardinal number.