Concept explainers
(a) Show that the map
EXAMPLE 9 The interval
is an order-preserving bijective correspondence, as you can check. It is pictured in Figure 3.2.
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Topology
- A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.arrow_forward6. In Example 3 of section 3.1, find elements and of such that but . From Example 3 of section 3.1: and is a set of bijective functions defined on .arrow_forwardFor any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .arrow_forward
- In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .arrow_forwardProve that if f is a permutation on A, then (f1)1=f.arrow_forwardDescribe 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 ].arrow_forward
- 6. a. Give an example of mappings and , different from those in Example , where is one-to-one, is onto, and is not one-to-one. b. Give an example of mappings and , different from Example , where is one-to-one, is onto, and is not onto.arrow_forwardIn Example 3 of Section 3.1, find all elements a of S(A) such that a2=e. From Example 3 of section 3.1: A=1,2,3 and S(A) is a set of bijective functions defined on A.arrow_forwardIn Exercises 1324, prove the statements concerning the relation on the set Z of all integers. If 0xy, then x2y2.arrow_forward
- 7. a. Give an example of mappings and , where is onto, is one-to-one, and is not one-to-one. b. Give an example of mappings and , different from example , where is onto, is one-to-one, and is not onto.arrow_forwardComplete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .arrow_forward
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