How to Find a Cartesian Product
- 1. Cartesian Product Definition: Given 2 sets A and B, the Cartesian Product is the set of all unique ordered pairs using one element from Set A and one element from Set B. The Cartesian Product is denoted as A x B A x B = { (a, b) | a ∈ A and b ∈ B } 1
- 2. Cartesian Product Formula Take 2 sets, A and B. The cardinality of a set equals the number of elements in the set, denoted |A|. The number of ordered pairs (cardinality) in the Cartesian Product equal |A x B| = |A| * |B| Each ordered pair is denote as {ai, bj} where… ● i is the ith element in Set A, j is the jth element in Set B 2
- 3. Cartesian Product Ordered Pair Grid A = {a1, a2}, B = {b1, b2, b3} |A x B| = |A| * |B| |A x B| = 2 * 3 = 6 3 b1 b2 b3 a1 (a1, b1) (a1, b2) (a1, b3) a2 (a2, b1) (a2, b2) (a2, b3)
- 4. Cartesian Product Ordered Pair Final Answer A x B = ((a1, b1), (a1, b2), (a1, b3), (a2, b1), (a2, b2), (a2, b3)} A = {a1, a2}, B = {b1, b2, b3} 4 b1 b2 b3 a1 (a1, b1) (a1, b2) (a1, b3) a2 (a2, b1) (a2, b2) (a2, b3)
- 5. Cartesian Product Example A = {1, 3}, B = {2, 4, 6} |A| = 2, |B| = 3 → |A x B| = 2 * 3 = 6 The Cartesian Product A x B contains 2 * 3 = 6 ordered pairs (a1, b1), (a1, b2), (a1, b3) = (1, 2), (1, 4), (1, 6) (a2, b1), (a2, b2), (a2, b3) = (3, 2), (3, 4), (3, 6) 5
- 6. Cartesian Product Example Ordered Pair Grid A = {1, 3}, B = {2, 4, 6} (a1, b1), (a1, b2), (a1, b3) = (1, 2), (1, 4), (1, 6) (a2, b1), (a2, b2), (a2, b3) = (3, 2), (3, 4), (3, 6) 6 b1 b2 b3 a1 (1, 2) (1, 4) (1, 6) a2 (3, 2) (3, 4) (3, 6)
- 7. Cartesian Product Example Final Answer A = {1, 3}, B = {2, 4, 6} A x B = {(1, 2), (1, 4), (1, 6), (3, 2), (3, 4), (3, 6)} 7 b1 b2 b3 a1 (1, 2) (1, 4) (1, 6) a2 (3, 2) (3, 4) (3, 6)