For a relation to be an equivalence relation on X, for any a, b and c in X the

following must hold:

(i) Reflexivity:a~a

(ii) Symmetry: ifa~bthenb~a

(iii) Transitivity: ifa~bandb~cthena~c.

1) X = R, let a, b and c be in R.

Clearly aRa <-> a-a is an integer but a-a=0 which is an integer so (i) holds

Also if a-b is an integer then so is b-a, so (ii) holds

Now if a-b is an integer, and so is b-c then:

a-c = (a-b) + (b-c)

is the sum of two integers and so is an iteger so aRc, so (iii) holds

Hence we conclude that R is an equivalence relation on R.

2) Same as above just go through checking reflexivity, symmetry and

transitivity if the hold the R is an equivelence relation if not it is not

RonL