Use example 3.2.2 to solve the exercises below:

Example 3.2.2. Let R be the relation on the set of real numbers R in Example

1. Prove that if xRx and yRy, then (x + y)R(x + y).

Proof:

Suppose xRx and yRy. In order to show that (x+y)R(x +y), we must show that (x + y) (x + y) is an integer. Since (x+y)(x +y)=(xx)+(yy), and since each of xx and yy is an integer (by definition of R), (xx)+(yy) is an integer. Thus, (x + y)R(x + y).

Exercise 3.2.1. In the example above, show that it is possible to have xRx and yRy, but (xy) R(xy).

Exercise 3.2.2. Let V be the set of vertices of a simple graph G. Define a relation R on V by vRw iff v is adjacent to w. Prove or disprove: R is an equivalence relation on V .