Basic Properties
Let A, B, C, and D be sets.
The Cartesian product A × B is not commutative,
because the ordered pairs are reversed except if at least one condition is satisfied:
- A is equal to B, or
- A or B is an empty set.
For example:
- A = {1,2}; B = {3,4}
- A × B = {1,2} × {3,4} = {(1,3), (1,4), (2,3), (2,4)}
- B × A = {3,4} × {1,2} = {(3,1), (3,2), (4,1), (4,2)}
- A = B = {1,2}
- A × B = B × A = {1,2} × {1,2} = {(1,1), (1,2), (2,1), (2,2)}
- A = {1,2}; B = ∅
- A × B = {1,2} × ∅ = ∅
- B × A = ∅ × {1,2} = ∅
Strictly speaking, the Cartesian product is not associative (unless the above condition occurs).
Read more about this topic: Cartesian Product
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