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.

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.

If A and B are sets and every element of A is also an element of B, then:

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:

We can prove the statement AB by applying a proof technique known as the element argument[5]:

The validity of this technique can be seen as a consequence of Universal generalization: the technique shows cAcB for an arbitrarily chosen element c. Universal generalisation then implies x(xAxB), which is equivalent to AB, as stated above.

Some authors use the symbols ⊂ and ⊃ to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇.[6] For example, for these authors, it is true of every set A that AA.

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 ⊋.[7][1] This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if xy, then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if AB, then A may or may not equal B, but if AB, then A definitely does not equal B.