Please do problem 9 and write a detailed proof when doing (a) 9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prove thatis an...

Question

Question

Please do problem 9 and write a detailed proof when doing (a)

9. Letbe the relation on the set of non-zero real numbers defined as follows: for r, y E R [0), x~ylf and only if-EQ (a) Prov

Answer 1

Answer #1

a)

For each non zero real number

x/x=1 which is rational

Hence relation is reflexive

let, x/y be rational then y/x is rational hence relation is symmetric

Let, x/y and y/z be rational

Product of rationals is rational hence

(x/y)(y/z)=x/z is rational

Hence relation is transitive

Hence relation is equivalence relation

b)

This equivalence class contains non zero real numbers x so that

$x/\pi ,\pi/x\textnormal{ is rational }$

Hence the real numbers is this equivalence class are

$\{q\pi ,q/\pi|q\in \mathbb{Q}\}$

Answer 2

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