In each casedetermine whether we have defined an equivalence

In each case,determine whether we have defined an equivalence relation or not. Give reasons for your answers.

Define three relations on R by x similar to y if and only if there exists n in [n,n+2) In each case,determine whether we have defined an equivalence relation or not. Give reasons for your answers. in [n, n+1) c) x,y in [n, n+1] b) x,y in Z such that a) x,y

Solution

a) does not exists such relations.

REASON : any two integers are related. as let n is a integer then n ~ (n+1) and (n+1) ~ (n+2) and so on. so only if condition does not hold.

b) [x] = the greatest integer function is the requard function.

REASON : x = y if and only if there exists integer a n such that n <= x,y < n+1.

c) does not exists such function.

REASON : any two integers are relsted by the same logic in a)