Let A, B, and C be sets (a) Prove that ACB iff A – B = Ø. (b) Prove that if A C BUCand ANB = Ø, then A C C. (c) Prove that (A – B) – C = (A – C) – (B – C).

Label each of the following statement as either true or false. 9. If ab and ba then a=b.

Show that if ax2+bx+c=0 for all x, then a=b=c=0.

6. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b.

For what values of c is c(1,2,3)=1?

7. For the given subsets and of Z, let and determine whether is onto and whether it is one-to-one. Justify all negative answers. a. b. c. d.

11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.

For the given subsets A and B of Z, let f(x)=2x and determine whether f:AB is onto and whether it is one-to-one. Justify all negative answers. a. A=Z+,B=Z b. A=Z+,B=Z+E

Let and . Prove or disprove that .