Browse Questions

# 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

(i) Since, a – a = 0 $\in$ Z, if follows that (a, a) $\in$ R.
(ii) (a,b) $\in$ R implies that a – b $\in$ 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 – c $\in$ Z. So, a – c = (a – b) + (b – c) $\in$ Z. Therefore, (a,c) $\in$ R