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Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
- 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.arrow_forwardProve that if m0 and (a,b) exists, then (ma,mb)=m(a,b).arrow_forwardMust two different points be collinear? Must three or more points be collinear? Can three or more points be collinear?arrow_forward
- Express (AB)(AB) in terms of unions and intersections that involve A,A,B,andBarrow_forwardIn 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 PorQandRarrow_forwardProve that the cancellation law for multiplication holds in Z. That is, if xy=xz and x0, then y=z.arrow_forward
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