A×A

A×A denotes the Cartesian product of a set A with itself. It is the set of all ordered pairs (a1, a2) where a1 ∈ A and a2 ∈ A. In many contexts A×A is also written as A^2, reflecting its two-coordinate structure. When A is a subset of the real numbers, A×A can be viewed as a subset of the plane R^2, consisting of all points whose coordinates lie in A.

Cardinality and examples. The size of A×A is the product of the sizes of the factors: |A×A| = |A| × |A|. If A is finite with n elements, then A×A contains n^2 elements; for example, if A = {1, 2, 3}, then A×A consists of nine pairs such as (1,1), (1,2), (1,3), (2,1), and so on. If A is infinite, then |A×A| = |A|; in particular, for countably infinite A, A×A is also countably infinite.

Structure and operations. Each element of A×A is a 2-tuple, and the set can be used to index or represent two-dimensional data, grids, or coordinate points when A ⊆ R. If A has its own binary operation, A×A naturally inherits a componentwise (direct product) operation: (a1,b1) ⋅ (a2,b2) = (a1⋅a2, b1⋅b2). The projection maps π1(a1,a2) = a1 and π2(a1,a2) = a2 extract the coordinates.

Related notions. A×A is a special case of the Cartesian product A×B. In many mathematical contexts A^2 is used interchangeably with A×A, by identifying A^2 with the set of functions from a two-element index set to A, which is naturally in bijection with A×A.