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- #1

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.

- Thread starter henry1
- Start date

- Thread starter
- #1

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.

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- #2

- Feb 15, 2012

- 1,967

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.