Cartesian Product - Basic Properties

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).