Proof by cases is a strategy requires a very specific setup but makes an otherwise difficult proof rather trivial.
- Proof by cases requires three statements: (1) P => R, (2) Q => R, and (3) P v Q. If you have these three, you can apply constructive dilemma and idempotence to conclude R.
- The process is called “proof by cases” because the disjunction P v Q means there are two cases to consider: P is true or Q is true. (The situation where both are true is superfluous.)
- The strategy works because the two implications say that in either case R is true, and we know that at least one of the cases must be true.
- For example, suppose you know: (1) if you are Tim Cook, you are rich; (2) if you are the CEO of a Fortune 500 company, you are rich; and (3) you are Tim Cook or you are the CEO of a Fortune 500 company. In either of those cases, you are rich. Thus, from that setup, we can conclude that you are rich.
- Most of the real work in a proof by cases is establishing the two implications. They usually will not be given to you. Instead, the majority of your proof will be dedicated to using conditional proofs to establish them.