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Equivalency relations

henry1

New member
Aug 25, 2013
1
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

Administrator
Staff member
Feb 24, 2012
13,775
Can you show what you have tried so far so our helpers can see where you are stuck or may be going astray?
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
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.