Subset Explained

In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide. The relationship of one set being a subset of another is called inclusion.

Definitions

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

• A is a subset of (or is included in) B, denoted by

,

or equivalently

• B is a superset of (or includes) 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 not contained in A), then

• A is also a proper (or strict) subset of B; this is written as

or equivalently

• B is a proper superset of A; this is written as

For any set S, the inclusion relation ⊆ is a partial order on the set 2^S of all subsets of S (the power set of S).

The symbols ⊂ and ⊃

Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So for example, for these authors, it is true of every set A that A ⊂ A.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively, in place of

and This usage makes ⊆ and ⊂ analogous to ≤ and <. For example, if x ≤ y then x may be equal to y, or maybe not, but if x < y, then x definitely does not equal y, but is strictly less than y. Similarly, using the "⊂ means proper subset" convention, if A ⊆ B, then A may or may not be equal to B, but if A ⊂ B, then A is definitely not equal to B.

Examples

• The set