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. There are no integers x and y such that 30x + 21y = 1. b. Ifn and m are integers such that n² + m² is odd, then m is odd or n is odd. c. For all integers x, y, and z, if y + z is divisible by x and y is divisible by x, then z is divisible by x.
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. There are no integers x and y such that 30x + 21y = 1. b. Ifn and m are integers such that n² + m² is odd, then m is odd or n is odd. c. For all integers x, y, and z, if y + z is divisible by x and y is divisible by x, then z is divisible by x.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterA: Appendix
SectionA.1: Algebraic Expressions
Problem 2E
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Solve a, b, c please
![You must write down all proofs in acceptable mathematical language: 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. See
lecture slides for examples.
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 r is rational if and only if there exist integers a and b with b + 0 such that
r= a/b.
For any real number x, the absolute value of x, denoted |x], is defined as follows:
{ x if x 20
|x|
l-x if x < 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. There are no integers x and y such that 30x + 21y = 1.
b. If n and m are integers such that n? + m² is odd, then m is odd or n is odd.
c. For all integers x, y, and z, if y + z is divisible by x and y is divisible by x, then z is
divisible by x.
d. The product of any non-zero rational number and any irrational number is irrational.
e. For all real numbers x and y, lx – y| = |y – x|.
f. Any two consecutive integers have opposite parity.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F672bf286-8abe-4b07-9ca1-0d5b2612956c%2Fba5f38f6-5cbe-484e-8ea9-9f0aa4e21a96%2Fbho252_processed.png&w=3840&q=75)
Transcribed Image Text:You must write down all proofs in acceptable mathematical language: 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. See
lecture slides for examples.
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 r is rational if and only if there exist integers a and b with b + 0 such that
r= a/b.
For any real number x, the absolute value of x, denoted |x], is defined as follows:
{ x if x 20
|x|
l-x if x < 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. There are no integers x and y such that 30x + 21y = 1.
b. If n and m are integers such that n? + m² is odd, then m is odd or n is odd.
c. For all integers x, y, and z, if y + z is divisible by x and y is divisible by x, then z is
divisible by x.
d. The product of any non-zero rational number and any irrational number is irrational.
e. For all real numbers x and y, lx – y| = |y – x|.
f. Any two consecutive integers have opposite parity.
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