Part I: Constructing proofs.

You must write down all proofs in acceptable mathematicallanguage: make sure you mark the beginning and end of the proof,define all variables, use complete, grammatically correctsentences, and give a justification for each assertion (e.g., bydefinition of…).

Definitions:

• An integer ? is even if and only if there exists an integer ?such that ? = 2?.

• An integer ? is odd if and only if there exists an integer ?such that ? = 2? + 1.

• Two integers have the same parity when they are both even orwhen they are both odd. Two integers have opposite parity when oneis even and the other one is odd.

• An integer ? is divisible by an integer ? with ? ≠ 0, denoted? | ?, if and only if there exists an integer ? such that ? =??.

• A real number ? is rational if and only if there existintegers ? and ? with ? ≠ 0 such that ? = ?/?.

• For any real number ?, the absolute value of ?, denoted |?|,is defined as follows: |?| = { ? if ? ≥ 0; −? if ? < 0 4.

Prove each of the following statements using a direct proof, aproof by contrapositive, a proof by contradiction, or a proof bycases. For each statement, indicate which proof method you used, aswell as the assumptions (what you suppose) and the conclusions(what you need to show) of the proof.

a. If ? is divisible by ? and ? is divisible by ?, where ?, ?,and ? are positive integers, then ? +? is divisible by ?.

b. The difference of any rational number and any irrationalnumber is irrational.

c. There is no integer that is both even and odd.

d. Any two consecutive integers have opposite parity.

e. For all real numbers ? and ?, ???(?, ?) = ?+?−|?−?| ? and???(?, ?) = ?+?+|?−?| ? .