Set Algebra Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

This discussion is an opportunity to explore the use of algebraic rules about set operations in proving relationships between sets. In particular, it investigates how set difference does or doesn’t distribute over other operations. Think about the following, and post any ideas, questions, etc. that you have. Complete solutions not required, of course, just ideas you or others can later build on.

Suppose A, B, and C are sets, all subsets of some universal set U, and consider the expressions (A ∪ B) - C and (A ∩ B) - C. Do you think set difference distributes over union and intersection in these expressions, i.e., that the first is equal to (A - C) ∪ (B - C) and the second to (A - C) ∩ (B - C)? Can you give examples to support your beliefs? Can you give proofs?

What if the difference is on the other side, i.e., C - (A ∪ B) and C - (A ∩ B)? Do difference and union or intersection interact the same way in this context as they did above? Can you illustrate with examples? Can you prove any conjectures you form?