For each of the statements below (which all happen to be false), look atthe “proofs” that have been provided and explain in one or two sentences what error has been made by the author. (a) Statement: The sum of any three odd integers equals 3p for some integer p."Proof": Suppose x, y and z are any three odd integers. By definition of odd,x = 2k +1 for some integer k, y = 2k +1 for some integer k, and z = 2k +1 forsome integer k.Now, r+ y +z = (2k + 1) + (2k + 1) + (2k + 1) (by substitution)(by algebra)(by algebra)= 6k + 3= 3(2k + 1)= 3pwhere we're letting p = 2k + 1, which is an integer by the closure properties ofintegers. We have shown that, for any odd integers r, y and z, r+ y + z = 3pfor some integer P, which completes the proof. I(b) Statement: For all integers m and n, if m +n is even, then m is even and n iseven."Proof": Let m and n be any even integers. Since m is even, m = 2p for someinteger p. Since n is even, n = 2q for some integer q.Therefore,m+n = 2p + 2q(by substitution)= 2(p+ q) (by algebra)= 2kwhere k = p+ q is an integer since it is the sum of two integers, p and q.Therefore, m + n is even, by definition of even. This completes the proof. I (c) Statement: For all odd integers m and n, " is an integer."Proof": Let m = 7 and let n = 1. 7 is odd because 7 = 2(3) + 1, and 3 is aninteger. 1 is odd because 1 = 2(0) + 1, and 0 is an integer.We wish to prove that " is an integer.m².6.m-1 = 4 (by substitution)(by algebra)= 8(by algebra)Since 8 is an integer, we have shown what we were required to show.

Answered: For each of the statements below (which…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

For each of the statements below (which all happen to be false), look at
the “proofs” that have been provided and explain in one or two sentences what error has been made by the author.

(a) Statement: The sum of any three odd integers equals 3p for some integer p.
"Proof": Suppose x, y and z are any three odd integers. By definition of odd,
x = 2k +1 for some integer k, y = 2k +1 for some integer k, and z = 2k +1 for
some integer k.
Now, r+ y +z = (2k + 1) + (2k + 1) + (2k + 1) (by substitution)
(by algebra)
(by algebra)
= 6k + 3
= 3(2k + 1)
= 3p
where we're letting p = 2k + 1, which is an integer by the closure properties of
integers. We have shown that, for any odd integers r, y and z, r+ y + z = 3p
for some integer P, which completes the proof. I
(b) Statement: For all integers m and n, if m +n is even, then m is even and n is
even.
"Proof": Let m and n be any even integers. Since m is even, m = 2p for some
integer p. Since n is even, n = 2q for some integer q.
Therefore,
m+n = 2p + 2q
(by substitution)
= 2(p+ q) (by algebra)
= 2k
where k = p+ q is an integer since it is the sum of two integers, p and q.
Therefore, m + n is even, by definition of even. This completes the proof. I

(c) Statement: For all odd integers m and n, " is an integer.
"Proof": Let m = 7 and let n = 1. 7 is odd because 7 = 2(3) + 1, and 3 is an
integer. 1 is odd because 1 = 2(0) + 1, and 0 is an integer.
We wish to prove that " is an integer.
m².
6.
m-1 = 4 (by substitution)
(by algebra)
= 8
(by algebra)
Since 8 is an integer, we have shown what we were required to show.

Expert Solution

Sub-part (a)

$Statement : The sum of any three odd integers equals 3 p for some integer p . In the proof of this statement, the author has assumed x = y = z = 2 k + 1 for some integer k . This is just a specific case, and not a general case . A better choice for x, y and z would be x = 2 a + 1 for some integer a y = 2 b + 1 for some integer b z = 2 c + 1 for some integer c Now, x + y + z = (2 a + 1) + (2 b + 1) + (2 c + 1) (b y s u b s t i t u t i o n) = 2 (a + b + c) + 3 (b y a l g e b r a) = 2 (a + b + c + 1) + 1 (b y a l g e b r a) = 2 p + 1 where we' re letting p = a + b + c + 1, which is an integer . As 2 p + 1 is an odd integer, therefore all we can claim is that the sum of any three odd integers in an odd integer .$