A set A is a SUBSET of B (written A ⊆ B) if every element of A is also an element of B. Every set is trivially a subset of itself (A ⊆ A), and the empty set is a subset of every set. If A ⊆ B, B is called a SUPERSET of A (B ⊇ A). A is a PROPER SUBSET of B (written A ⊂ B) if A ⊆ B but A ≠ B — that is, B contains at least one extra element that A lacks. The POWER SET of A, written P(A), is the set of ALL subsets of A (including φ and A itself) — so every 'element' of a power set is itself a set. If n(A) = m, then the power set always has exactly n[P(A)] = 2^m elements, since each of A's m elements independently either is or isn't included in a given subset, giving 2×2×...×2 (m times) possible subsets. For example, A = {a,b} has power set P(A) = {φ, {a}, {b}, {a,b}}, with 2² = 4 elements. A UNIVERSAL SET (usually U or X) is simply whichever fixed 'background' set every set under discussion is agreed to be a subset of, for a given problem — e.g. the real numbers R often serve as the universal set when discussing N, Z, and Q together. The universal set is what makes the COMPLEMENT of a set meaningful (see the separate Complement concept).