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Cartesian Product of Two Sets


Let A and B be two sets. The Cartesian product or cross product of A and B, denoted by A × B, is the set of all ordered pairs (a, b) such that a A and b B.

That is, A × B = {(a, b) |a A, b B}.
Thus, {1, 2, 3} × {a, b} = {(1, a), (2, a), (3, a), (1, b), (2, b), (3, b)}

Note carefully that
  1. A ×B B × A unless B = A.
  2. If n(A) stands for the number of elements in A, then n(A × B) = n(B × A) = n(A) × n(B).
For instance, if n(A) = 3, n(B) = 5, then n(A × B) = n(B × A) = (3) (5) = 15.
  1. If either of A or B is a null set, then A × B is also a null set.
  2. If either of A or B is an infinite set, then A × B is also an infinite set.
  3. A ×A × A = {(a, b, c) | a, b, c A}. Here (a, b, c) is called an ordered triplet.




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