Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 24E: For any relation on the nonempty set, the inverse of is the relation defined by if and only if ....

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Let y={0,1,2} and 2={0,1} and define a relation r from a to b as follows: Given any (x,y)€ Y×Z. (x,y)€ R means that x+y/2 is an integer

1. State explicitly which ordered pairs are in YxZ and which are in R.

2. Is 1 R 0? Is 2 R 0? Is 2 R 1?

3. What are the domain are co-domain of R?

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Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
For 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 .
2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric or transitive. Justify your answers. a. if and only if . b. if and only if . c. if and only if for some in . d. if and only if . e. if and only if . f. if and only if . g. if and only if . h. if and only if . i. if and only if . j. if and only if . k. if and only if .

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0,1,2

Let y={0,1,2} and 2={0,1} and define a relation r from a to b as follows: Given any (x,y)€ Y×Z. (x,y)€ R means that x+y/2 is an integer​

Let y={0,1,2} and 2={0,1} and define a relation r from a to b as follows: Given any (x,y)€ Y×Z. (x,y)€ R means that x+y/2 is an integer