8 Consider the following binary relation: Vx, y E R, x Z y if x1 > Y1 (indepen-dent of x2 and y2) or if x1Yı and a2 > y2. These preferences are calledlexicographic.a) Show that this preference relation is complcte and transitive, but notcontinuous.b) Why is this preference relation not very interesting from an cconomic pointof view?

We will use the basic knowledge of relations to answer this question correctly and completely.…

Answered: Consider the following binary relation:…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 13E: 13. Consider the set of all nonempty subsets of . Determine whether the given relation on is...

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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.
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.
Prove the transitivity of strict preference relation.
Let RR be a binary relation on A={a,b,c}A={a,b,c} given by R={(a,a),(c,c),(a,b),(b,c),(b,a)}R={(a,a),(c,c),(a,b),(b,c),(b,a)}. Explain why: (a) RR is *not * reflexive. (b) RR is *not * symmetric. (c) RR is *not * transitive.
shows that there is no topology in X based on the familyB = {{a, b}. {a, b, d}, {b, d, c}}
Let S = {0, 1, 2, 4, 6}. Test the following binary relation on S for reflexivity, symmetry, antisymmetry, and transitivity: p = {(0, 1), (1, 2), (0, 2), (2, 0), (2, 1), (1, 0), (0, 0), (1, 1), (2, 2)}
Let A = {1,2,3,4}, B = {2,3,4,5}, and C = {2,4,6,8,10}. Suppose R is a 3-ary relation that is defined by the following rule: (a,b,c) is an element of R provided that a + 1 = b and b | c (meaning that b divides c with a remainder of zero), and where a ∈ A, b ∈ B, and c ∈ What triples are in the 3-ary relation R?
Compute the complement of relation R. R={(1, 1), (5, 1), (5, 5), (6, 1), (6, 6), (7, 1), (7, 7)}
Prove the second absorption law from Table 1 by showing that if A and B are sets, then A ∩ (A ∪ B) = A.
Consider the set of all members belonging to the same family and the binary relation defined as " to be an ancestor of " . 1. Is Rrational ? Explain 2. A binary relation is said asymmetric if XRy , then not yox . Is R asymmetric ? Explain
Define the relation∼ on Nby m ∼ n if and only if the sum of the distinct primes that divide m is the same as the sum of the primes that divide n. For example, 12 ∼ 5 since the sum of the primes that divide 12 ( 2 + 3 ) is the same as the sum of the primes that divide 5 ( 5 ). Is ∼ an equivalence relation? Explain how you know, either providing a counterexample or briefly (not a full proof--examples are fine) explaining how you know ∼ is reflexive, symmetric, and transitive. If ∼is an equivalence relation, find a few elements of the following equivalence classes:
Prove the domination laws in Table 1 by showing that a) A ∪ U = U. b) A ∩ ∅ = ∅.

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