Prove or disprove: If R and S are partial orderings on A, then R ∩ S is also a partial ordering on A.
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Prove or disprove: If R and S are partial orderings on A, then R ∩ S is also a partial ordering on A.
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- Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .
- In Exercises , prove the statements concerning the relation on the set of all integers. 14. If and , then .12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 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.In Exercises , prove the statements concerning the relation on the set of all integers. 17. If and , then .