Subset

Euler diagram showing
A is a subset of B (denoted ) and, conversely, B is a superset of A (denoted ).

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.

When quantified, is represented as [1]

One can prove the statement by applying a proof technique known as the element argument[2]:

Let sets A and B be given. To prove that

  1. suppose that a is a particular but arbitrarily chosen element of A
  2. 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.

  1. ^ Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5.
  2. ^ Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). p. 337. ISBN 978-0-495-39132-6.