Proving Equivalency Relations: Help from Henry

[henry1]

I'm having copious amounts of trouble with this question and an amount of help would really be appreciated.

Let S be the relation on the set of real numbers defined by

x S y iff x-y is an integer

1. prove that S is an equivalence relation on R.

2. Prove that if x S x' and y S y' then (x+y) S (x'+y').

Thanks,

Henry.

 

[MarkFL]

Can you show what you have tried so far so our helpers can see where you are stuck or may be going astray?

 

[Deveno]

My thoughts:

S is reflexive because 0 is an integer.

S is symmetric because -k is an integer whenever k is.

S is transitive, because the sum of two integers is another integer.

The second part of the problem is to show S is a congruence with respect to addition (of real numbers). This really just amounts to working through the definition of S:

Suppose x S x'. Then x - x' = k, for some integer k. Similarly, y S y' means y - y' = m, for some integer m.

Consequently:

(x + y) - (x' + y') = (x - x') + (y - y') = k + m, which is, of course, an integer.