Cartesian Product Calculator
Calculate the Cartesian product A x B of sets to get all ordered pairs.
Define Sets
|A| = 3
|B| = 2
Cardinality
|A × B| = |A| × |B| = 3 × 2 = 6
A × B = B × A?NO
Cartesian product is generally NOT commutative
A × B
6 ordered pairs
(1, a)(1, b)(2, a)(2, b)(3, a)(3, b)
B × A
6 ordered pairs
(a, 1)(a, 2)(a, 3)(b, 1)(b, 2)(b, 3)
A × A
9 ordered pairs
(1, 1)(1, 2)(1, 3)(2, 1)(2, 2)(2, 3)(3, 1)(3, 2)(3, 3)
About Cartesian Product
Definition
A × B = {(a, b) | a ∈ A and b ∈ B} - the set of all ordered pairs where the first element comes from A and second from B.
Properties
- Not commutative: A × B ≠ B × A (usually)
- |A × B| = |A| × |B|
- A × ∅ = ∅ × A = ∅