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.
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.
Chapter8: Sequences, Series,and Probability
Section8.6: Counting Principles
Problem 24E
Related questions
Question
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 is
odd 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. Two
integers 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 an
integer k such that n = dk.
A real number is rational if and only if there exist integers à and b with b + 0 such that
r = 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 ≥ 0
l-x if x < 0
|x|
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda22bb93-89e0-43b0-b495-0a6067b20da3%2Fea0e86db-b35d-4e74-8d03-5739c06d5cdc%2Fvt6wp8_processed.png&w=3840&q=75)
Transcribed Image Text:Definitions:
An integer n is even if and only if there exists an integer k such that n = 2k. An integer n is
odd 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. Two
integers 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 an
integer k such that n = dk.
A real number is rational if and only if there exist integers à and b with b + 0 such that
r = 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 ≥ 0
l-x if x < 0
|x|
=
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