#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) = { }.
To prove this using a proof by contradiction, we assume the negation of the statement we want to prove, and then show that it leads to a contradiction.
So, let's assume that the intersection of (A x B) and (C x D) is not empty, i.e., there exists an element in (A x B) intersection (C x D). Let's call this element "x".
By definition, (A x B) is the set of all ordered pairs (a, b), where a is an element of A, and b is an element of B. Similarly, (C x D) is the set of all ordered pairs (c, d), where c is an element of C, and d is an element of D.
Now, if "x" belongs to (A x B) intersection (C x D), it means that "x" is an ordered pair in both (A x B) and (C x D). Therefore, we can write "x" as (a, b) and also as (c, d), where a is in A, b is in B, c is in C, and d is in D.
Since (A intersection C) is an empty set, it implies that a belongs to A and c belongs to C, but a doesn't belong to C and c doesn't belong to A.
Now, a belongs to A and c belongs to C implies that (a, c) belongs to (A x C). Similarly, b belongs to B and d belongs to D implies that (b, d) belongs to (B x D).
So, if we consider the ordered pair (a, c) from (A x C) and the ordered pair (b, d) from (B x D), we can form a new ordered pair (a, d) by taking the first element from (a, c) and the second element from (b, d).
Now, let's check whether (a, d) belongs to (A x B) intersection (C x D). By definition, (a, d) belongs to (A x B) if and only if a belongs to A and d belongs to B. Similarly, (a, d) belongs to (C x D) if and only if a belongs to C and d belongs to D.
But we know that a belongs to A and d belongs to D, which means that (a, d) belongs to both (A x D) and (C x D). Therefore, (a, d) belongs to (A x D) intersection (C x D), which contradicts our assumption that (A x B) intersection (C x D) is empty.
Since our assumption led to a contradiction, our initial assumption must be false. Therefore, we can conclude that if A intersection C is an empty set, then (A x B) intersection (C x D) is also an empty set.