B×C

B×C denotes the Cartesian product of sets B and C. It is the set of all ordered pairs (b, c) where b belongs to B and c belongs to C.

Formally, B×C = { (b, c) | b ∈ B and c ∈ C }. The order of the factors matters: B×C and C×B are generally different sets. If either B or C is empty, then B×C is the empty set. There are natural projection maps π_B: B×C → B and π_C: B×C → C given by π_B(b, c) = b and π_C(b, c) = c, which “forget” the other component.

Cardinality and size: If B and C are finite with |B| = m and |C| = n, then |B×C| = m·n. For infinite sets, the cardinality of the product is the product of the cardinals |B×C| = |B|·|C|, and in many standard cases this equals max(|B|, |C|).

Examples help: if B = {1, 2} and C = {a, b, c}, then B×C = { (1, a), (1, b), (1, c), (2, a), (2, b), (2, c) }. The Cartesian product underpins many constructions in mathematics, including the domain for relations from B to C: a relation is a subset of B×C. It also generalizes to n-ary products B1×…×Bn and serves as a foundation for product topologies, database joins, and other combined-structure concepts.

Related topics include set-theoretic operations on Cartesian products, properties of projections, and the role of Cartesian products in enumeration and algebraic constructions.