Philosophy of basic set theory proofs involving "or". Hey! I'm working through an Introduction to Analysis text, and I'm currently on the first chapter, which covers set theory. In one of the end-of-chapter problems, I'm asked to prove a basic theorem which leads to the following statement: x is an element of A, and (x is an element of B or x is an element of C). My text (Maxwell Rosenlicht's Introduction to Analysis) lacks in the "example" department, and so for a little while, I wasn't sure how to handle this statement. I've been pondering this for a few days, and realized that "a or b" is true if: 1. a is true. b is false. 2. b is true. a is false. 3. both a and b are true. So, I realized that the only thing I could do was split the argument into these three possibilities. I then showed that for my particular proof, all three possibilities lead to the same statement. I have a quick question: Is it common practice in set theory to split "or" statements into the three possible statements which can make it true, and proceed in this way? It makes a lot of sense intuitively, but I've never seen it done in any professional papers. Is there any other way to handle such statements? Thanks.