Show that the relation R on ZxZ defined by (a, b) R (c, d) if and only if a + d = b + c is an equivalence relation.

Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .

Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.