Consider the following Useful FactTM: If A B and A C, then A (B C). The proof isnt too difficult. Consider any element x A. Since A B, we know that x B. Similarly, since A C, we know that x C. By definition of the set intersection, it follows that x (B C). This reasoning applies to any element x A weve just shown that any element in A is also an element of (B C). Therefore, A (B C). Q.E.D. The Useful Fact above gives us a technique to prove statements of the form A (B C). We can simply do two separate proofs to show that A B and A C. If we can show that both of these are true, then by the Useful Fact we can conclude that A (B C). Use this approach to prove A (B C) for the following sets. (a) (2 points) A = {10k 1 : k Z}, B = {2m + 1 : m Z}, C = {5n + 4 : n Z} Note: The notation {f(x) : x Z} means the set of all values that can be produced by substituting any integer x into f(x). For example, {2k + 1 : k Z} is another way of saying the set of all odd integers. (b) (2 points) A = {x R : x 2 < 4}, B = {x R : x < 2}, C = {x R : x > 2}