#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.
To complete the proof by contradiction, we need to argue why the assumption leads to a contradiction.
You correctly assume that (m, n) ∈ (A x B) ⋂ (C x D) implies (m, n) ∈ (A x B) and (m, n) ∈ (C x D). This means m ∈ A and n ∈ B, and also m ∈ C and n ∈ D.
However, you claim m ∈ A and m ∈ C, which leads to the assumption m ∈ (A⋂C). This assumption is incorrect because we know from the given statement that A ⋂ C = {}. Since m is an element of both A and C, it cannot be an element of the empty set. This is the contradiction.
To formalize the contradiction, you can say:
Since (m, n) ∈ (A x B) ⋂ (C x D), we have m ∈ (A ⋂ C), which contradicts the fact that A ⋂ C = {}. Therefore, our assumption (A x B) ⋂ (C x D) = { } is false.
Thus, by proof by contradiction, we conclude that (A x B) ⋂ (C x D) = { } is true.