You must write down all proofs in acceptable mathematical language: make sure you mark the beginning and end of the proof, define all variables, use complete, grammatically correct sentences, and give a justification for each assertion (e.g., by definition of…).
Prove each of the following statements using a direct proof, a proof by contrapositive, a proof by contradiction, or a proof by cases. For each statement, indicate which proof method you used, as well as the assumptions (what you suppose) and the conclusions (what you need to show) of the proof.
1. If ? is divisible by ? and ? is divisible by ?, where ?, ?, and ? are positive integers, then ? +? is divisible by ?
2. The difference of any rational number and any irrational number is irrational.
3. There is no integer that is both even and odd.
4. Any two consecutive integers have opposite parity.
5. For all real numbers ? and ?, ???(?, ?) = (?+?−|?−?|) / ? and ???(?, ?) = (?+?+|?−?|) / ? .