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Let R be a.relation from Q to Q defined ...

Let `R` be `a`.relation from `Q` to `Q` defined by `R={(a, b): a, b in Q` and `a-b in Z }.` Show that
i) `(a, a) in R` for all `a in Q`
ii) `(a, b) in R` implies that `(b, a) in R`
iii) `(a, b) in R` and `(b, c) in R` implies that `(a, c) in R .`

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i) Since, `a-a=0 in Z`, if follows that `(a, a) in R_i`
ii) `(a, b) in R` implies that `a-b cdot in Z`. So, `b-a in Z`. Therefore, `(b, a) in R`.
iii) `(a, b)` and `(b, c) in R` implies that `a-b in Z . b-d in Z . S o, a-c=(a-b)+(b-c) in Z` Therefore, `(a, c) in R`
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