1 votes 1 votes Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if x ≡ y (mod 7) Set Theory & Algebra set-theory&algebra discrete-mathematics engineering-mathematics set-theory + – Vicky rix asked Mar 15, 2017 Vicky rix 1.4k views answer comment Share Follow See all 0 reply Please log in or register to add a comment.
Best answer 1 votes 1 votes 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. Vicky rix answered Mar 15, 2017 selected Nov 5, 2017 by Manu Thakur Vicky rix comment Share Follow See 1 comment See all 1 1 comment reply reena_kandari commented Nov 5, 2017 reply Follow Share $7$mod $7$=$0$, how $(7,7)$ are related to each other. your way is correct but just a small confusion!! 0 votes 0 votes Please log in or register to add a comment.