In 19-31,(1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence dasses of each relation.

I is the relation defined on R as follows:

For every                     m / n                     ⇔     x − y         is     an integer .

To prove:

The given relation is an equivalence relation and also find the distinct equivalence classes of given relation.

Given information:

I is the relation defined on ⁒ as follows:

For all x, y ∈ R, x I y ⇔ x - y is an integer.

Calculation:

I={(m,n)∈⁒×⁒|m−n is an integer}

To prove: R is an equivalent relation.

Reflexive:

Let x∈⁒

x−x=0

x−x is an integer, since 0 is an integer and thus (x,x)∈I

Symmetric:

Let us assume that (x,y)∈I.

By the definition of I :

x−y is an integer.

However the negation of an integer is also an integer.

−(x−y) is an integer.

Use distributive property,

y−x is an integer.

By the definition of I :

(y,x)∈I

Since (x,y)∈I implies (y,x)∈I, I is symmetric.

Transitive:

Let us assume that (x,y)∈I and (y,z)∈I.

By the definition of I :

(x−y) is an integer