Theorem 5.8: If S is any set, then

Proof: Suppose that { } x S is not empty. Then there exists an element (r, s) in { } x S. Then, by the definition of the Cartesian product, Definition 1.9, r would be an element of the empty set, which is impossible by Definition 3.1.

Similarly if S x { } is not empty, there exists an element (s , t ) in S x { }. Then t would be an element of the empty set, which is impossible by Definition 3.1.