(12)Prove that if x EZ and r² is even, then x is even.(b) State and prove an analogous statement to (a) about odd numbers.

Answered: (12) (a) Prove that if x € Z and r² is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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The set S contains some real numbers, according to the following three rules.(i) 1/1 is in S.(ii) If a/b is in S, where a/b is written in lowest terms (that is, a and b have highest common factor 1), then b/2ais in S. (iii) If a/b and c/d are in S, where they are written in lowest terms, then a+c/b+d is in S.These rules are exhaustive: if these rules do not imply that a number is in S, then that number is not in S. Can you describe which numbers are in S? For example, by (i), 1/1 is in S. By (ii), since 1/1 is in S, 1/2·1is in S. Since both 1/1 and 1/2 are in S, (iii) tells us 1+1/1+2 is in S. What I have so far: Claim: Set S in contained in interval [½, 1] for a/b where 0<a≤b≤2a The reason is that 1/1 has this form and transformations preserve the property of being in this interval If a≤b≤2a, then b/2a obeys the requirement, since b≤2a≤2b And if a/b and c/d obey the requirement, then so does (a+c)/(b+d), since a+c≤b+d≤2a+2c=2(a+c) However, I feel there is still more to this…
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(12) (a) Prove that if x € Z and r² is even, then x is even. (b) State and prove an analogous statement to (a) about odd numbers.