Question

#2: Use a proof by contradiction to prove:

For all sets A, B, C, D if A intersection C= { },
then (A x B) intersection (C x D) = { }.

This is my proof so far, but I don't know how to solve it fully.

Assume that (A x B) ⋂ (C x D) = { }. Then (m, n) ∈ (A x B) ⋂ (C x D). This means that (m, n) ∈ (A x B) and (m, n) ∈ (C x D). Thus, m∈A and n∈B. Also, m∈C and n∈D. Thus, m∈A and m∈C. Therefore, m ∈ (A⋂C) ... contradiction ... (A x B) ⋂ (C x D) = { } is false.
Therefore, (A x B) ⋂ (C x D) = { } is true.