Prove that: If f E Ra, b), then |fI E Rla, b) with

Show that u and v in the figure at the right are equivalent.

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.

Prove that if , , and , then .

To prove a theorem of the form "If P, then Q" by the indirect method, the first line of the proof should read: Suppose that ___________ is true.

Must two different points be collinear? Must three or more points be collinear? Can three or more points be collinear?

Does it follow from Exercise 30 that ADE is also congruent to CBE? What can you conclude regarding AE and CE in the drawing? What can you conclude regarding DEandBE?

In Exercises 55 and 56, P is a true statement, while Q and R are false statements. Classify each of the following statements as true or false. a PandQorR b PorQandR

If a0 and ab, then prove or disprove that (a,b)=a.

True or false Label each of the following statement as either true or false. Let and , then .