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we say that x $\equiv$ y(mod n) iff n divides (x-y) (i.e) (x-y) is a multiple of 7.

                     here relation is x $\equiv$ y(mod 7) which means (x-y) = 7m where m is some integer.

                     If (x-y) = 7m then (y-x) = -7m.so i can also write as y $\equiv$ x(mod 7).

                     So if (x,y) $\in$ R , then (y,x) $\in$ R.So this relation is symmetric. Also this shows this relation is not anti-symmetric.

This relation is reflexive because x $\equiv$ x(mod n) because (x-x) = 0 which is divisible by 7.

This relation is transitive because

                 if (x,y) $\in$ R, then x $\equiv$ y(mod 7) (i.e) (x-y) = 7m ---->1

                 if (y,z) $\in$ R, then y $\equiv$ z(mod 7) (i.e) (y-z) = 7p.---->2

      1) + 2) will give (x-z) = 7m+7p =7(m+p).so since (x-z) is also a multiple of 7, x $\equiv$ z(mod 7), which means (x,z) $\in$ R.

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