Below is the Hasse diagram for a partial order relation Ron the set {a, b, C, d, e}.edbeaAre the following statements true or false?True O 1. (a, b) E RTrue O 2. (b, c) E RFalse O 3. (a, d) E RFalse O 4. (c, e) E RFalse O 5. (e, a) E RTrue O 6. (a, a) E R

Introduction: Hasse diagram is very significant in graphing a relation. In this diagram, the…

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 14E: In each of the following parts, a relation is defined on the set of all human beings. Determine...

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In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.
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 each of the following parts, a relation R is defined on the power set (A) of the nonempty set A. Determine in each case whether R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if xy. b. xRy if and only if xy.

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Below is the Hasse diagram for a partial order relation R on the set {a, b, c, d, e}. e d be C Are the following statements true or false? True O 1. (a, b) E R True o 2. (b, c) E R False O 3. (a, d) E R False O 4. (c, e) E R False O 5. (e, a) E R True O 6. (a, a) E R