Prove each of the following statements using a direct proof, a proof by contrapositive, or a proof by contradiction (and cases when needed). 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 any integers x and y, if x + y > 50, then x > 20 or y > 30.B.) For any integers x, y, and z with x ≠ y, if z is divisible by x − y, then z is divisible by y − x.C.) The difference of any rational number and any irrational number is irrational. Definitions:An integer n is even if and only if there exists an integer k such that n = 2k. An integer n isodd if and only if there exists an integer k such that n = 2k + 1.●●●Two integers have the same parity when they are both even or when they are both odd. Twointegers have opposite parity when one is even and the other one is odd.An integer n is divisible by an integer d with d 0, denoted d [ n, if and only if there exists aninteger k such that n = dk.A real number is rational if and only if there exist integers à and b with b + 0 such thatr = a/b.For any real numbers x and y, the average of x and y is given by (x + y)/2.For any real number x, the absolute value of x, denoted [x], is defined as follows:[x ifx ≥ 0l-x if x < 0|x|=

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Answered: A.) For any integers x and y, if x + y…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove each of the following statements using a direct proof, a proof by contrapositive, or a proof by contradiction (and cases when needed). 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 any integers x and y, if x + y > 50, then x > 20 or y > 30.
B.) For any integers x, y, and z with x ≠ y, if z is divisible by x − y, then z is divisible by y − x.
C.) The difference of any rational number and any irrational number is irrational.