P×M

P×M denotes the Cartesian product of two sets P and M. It is the set of all ordered pairs (p, m) where p is a member of P and m is a member of M. The order matters, so (p, m) generally differs from (m, p).

Canonical projections π_P and π_M map P×M to P and M, respectively, by π_P(p, m) = p and π_M(p, m) = m. These projections are useful for extracting components of elements in the product.

Example: If P = {a, b} and M = {1, 2, 3}, then P×M = {(a,1), (a,2), (a,3), (b,1), (b,2), (b,3)}.

Cardinality: If P and M are finite with sizes |P| and |M|, then |P×M| = |P|·|M|. If either set is infinite, the cardinality of P×M is the product of their cardinalities, and in many common cases this equals or exceeds the larger of the two.

Algebraic context: In algebra, P×M often denotes the direct product of two structures, such as groups or rings, where the operation is defined componentwise. For example, in groups, (p1, m1)·(p2, m2) = (p1·p2, m1·m2). The direct product satisfies a universal property with projection maps, mirroring the Cartesian product’s universal property in the category of sets.

Variants and related concepts: For more factors, P×M×N denotes the Cartesian product of three sets; in topology and category theory, the Cartesian product serves as a product object, with similar componentwise structure in algebraic settings.