Let A be the set of integers, and define a relation R on A by mRn if and only if m − n is a multiple of 5. Show that R is an equivalence relation, and indicate the equivalence classes of R.

Answer 1

R is reflexive. m R m because m-m = 0 which is a multiple of 5

R is symmetric. m R n then m-n = 5k where k is an integer. n-m = -5k = 5 (-k) where -k is an integer.

So if mRn then nRm

R is transitive. mRn and nRp then m-n = 5k, n-p=5l, k and l are integers

m-p = m-n+n-p = 5k + 5l = 5 (k+l). k+l is an integer so mRp.

5 equivalence classes. { x : x = 5k, k an integer } { x+1 : x = 5k } {x+2 : x = 5k } { x+3 : x = 5k} {x+4 : x = 5k}