Question

Write down all proofs in acceptable mathematical language: make sure that you mark the beginning and end of the proof, state every assumption, define every variable, give a justification for every assertion (e.g., by definition of…), and use complete, grammatically correct sentences.

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 or when they are both odd. Two integers have opposite parity when one is 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 exist integers ? and ? with ? ≠ 0 such that ? = ?/?.

• For any real number ?, the absolute value of ?, denoted |?|, is defined as follows: |?| = { ? if ? ≥ 0 −? if ? < 0 2.

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.

a. For all integers ?, ?, and ?, if ?? is not divisible by ?, then ? is not divisible by ?.

b. For all positive integers ?, ?, and ?, if ? is divisible by ? and ? is divisible by ?, then ? + ? is divisible by ?.

c. The difference of any rational number and any irrational number is irrational

d. There are no integers ? and ? such that ???+ ?? = ?.

e. Any two consecutive integers have opposite parity.

f. For all real numbers ? and ?, ???(?, ?) = ?+?−|?−?| ? and ???(?, ?) = ?+?+|?−?| ? , where ???(?, ?) and ???(?, ?) denote the minimum and the maximum of ? and ?, respectively.

Answer 1