#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) = { }.
So do I assume the Hypothesis is false or do I assume the conclusion is false to do a proof by contradiction?
To use a proof by contradiction, you assume that the hypothesis (the statement you want to prove) is true and then show that this leads to a contradiction, proving that the hypothesis must be false.
In this case, the hypothesis is "A ∩ C = {}" and the conclusion is "(A x B) ∩ (C x D) = {}". So, to prove this using a proof by contradiction, you assume that the hypothesis is true and the conclusion is false. In other words, you assume that A ∩ C = {} is true, but (A x B) ∩ (C x D) ≠ {}.
Next, you need to show that this assumption leads to a contradiction. To do this, suppose (A x B) ∩ (C x D) ≠ {} under the assumption that A ∩ C = {}. By the definition of set intersection, this means that there exists an element in (A x B) ∩ (C x D). Let's call this element x.
Since x is in (A x B) ∩ (C x D), it must be in both (A x B) and (C x D). By the definition of Cartesian product, an element (a, b) is in A x B if and only if a is in A and b is in B. Similarly, an element (c, d) is in C x D if and only if c is in C and d is in D.
Now, consider x = (a, b) = (c, d), where a is in A, b is in B, c is in C, and d is in D. Since x is in (A x B) and (C x D), this means a is in A and c is in C. However, since A ∩ C = {}, this implies that a is not in A ∩ C (which is an empty set) and therefore a is not equal to c, leading to a contradiction.
As a result, since assuming the hypothesis leads to a contradiction, we can conclude that the hypothesis must be false. Therefore, we have proven by contradiction that for all sets A, B, C, D, if A ∩ C = {}, then (A x B) ∩ (C x D) = {}.