6.2.36 Prove that, for all sets A and B, a. (A B)U (BA) U(ANB) = AUB b. The sets (AB), (BA), and (ANB) are mutually disjoint.

Determine whether the set S={1,x2,2+x2} spans P2.

[Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and multiplication are defined as follows: if and only if and in , Given that is a ring, determine whether is commutative and whether has a unity. Justify your decisions. [Type here]

In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .

32. Prove that if is positive and is positive, then is positive.

31. Prove that if is positive and is negative, then is negative.

For each of the following relations R defined on the set A of all triangles in a plane, determine whether R is reflexive, symmetric, or transitive. Justify your answers. a. aRb if and only if a is similar to b. b. aRb if and only if a is congruent to b.

In Exercises 13-24, prove the statements concerning the relation on the set of all integers. 19. If and, then.

Label each of the following statements as either true or false. Let f:AB. Then f(A)=B for all nonempty sets A and B.

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