Kenneth Rosen Edition 6th Exercise 7.5 Question 57 (Page No. 509)

Question

Consider the equivalence relation R = $\{(x,y) \, | \, x-y \,is\,an\,integer\}$

(b) What is the equivalence class of 1/2 for this equivalence relation?

Comments

$\{...-2.5,-1.5,.5, 1.5........\}?$

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Seems to be correct because answer given is $\{n+\frac{1}{2}.n \, | \, n \in Z\}$

but how can it contain some fractional elements because the relation is on Z?

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How relation is on $Z$ ? It must be on $R$ because equivalence class of $\frac{1}{2}$ is asked.

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@Ayush, Given answer is $\{n+\frac{1}{2}.n|n∈Z\} \ or \ \{n+\frac{1}{2}|n∈Z\}?$

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Sorry soumya, it was a typo. Answer is the latter expression in your comment above

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So for this to be possible, does this relation R needs to be necessarily defined on real numbers and not integers?

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Yes. Otherwise, if it were defined on $Z$ then it would be empty.