Let A ⊆ R be nonempty and bounded above, and let S be the union of all A ∈ A. (a) First, prove that S ∈ R by showing that it is a cut. (b) Now, show that S is the least upper bound for A. This finishes the proof that R is complete. Notice that we could have proved that least upper bounds exist immediately after defining the ordering on R, but saving it for last gives it the privileged place in the argument it deserves. There is, however, still one loose end to sew up. The statement of Theorem 8.6.1 mentions that our complete ordered field contains Q as a subfield. This is a slight abuse of language. What it should say is that R contains a subfield that looks and acts exactly like Q.

Let A ⊆ R be nonempty and bounded above, and let S be the union of all A ∈ A. (a) First, prove that S ∈ R by showing that it is a cut. (b) Now, show that S is the least upper bound for A. This finishes the proof that R is complete. Notice that we could have proved that least upper bounds exist immediately after defining the ordering on R, but saving it for last gives it the privileged place in the argument it deserves. There is, however, still one loose end to sew up. The statement of Theorem 8.6.1 mentions that our complete ordered field contains Q as a subfield. This is a slight abuse of language. What it should say is that R contains a subfield that looks and acts exactly like Q.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 29E: 29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there...

See similar textbooks

Similar questions

29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .
Label each of the following statements as either true or false. Every upper bound of a nonempty set is a least upper bound.
Label each of the following statements as either true or false. If a nonempty set contains an upper bound, then a least upper bound must exist in .
Label each of the following statements as either true or false. Every least upper bound of a nonempty set S is an upper bound.
Label each of the following statements as either true or false. The least upper bound of a nonempty set S is unique.
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.
6. Prove that if is a permutation on , then is a permutation on .
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
Describe the kernel of epimorphism in Exercise 20. Consider the mapping :Z[ x ]Zk[ x ] defined by (a0+a1x++anxn)=[ a0 ]+[ a1 ]x++[ an ]xn, where [ ai ] denotes the congruence class of Zk that contains ai. Prove that is an epimorphism from Z[ x ] to Zk[ x ].
Prove that if f is a permutation on A, then (f1)1=f.
Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.