Part i 6. Suppose A is the set of counties in Leinster.(i) Give an example of a relation on A that is an equivalence relation. Prove that yourexample is an equivalence relation.(ii) Give an example of a relation on A that is not an equivalence relation. Prove thatyour example is not an equivalence relation.

Step 1: (i) Example of an Equivalence Relation on A Definition of an Equivalence RelationOne kind of…

Answered: 6. Suppose A is the set of counties in…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 11E: Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide...

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Similar questions

Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.
Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
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.
Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
Let A=R0, the set of all nonzero real numbers, and consider the following relations on AA. Decide in each case whether R is an equivalence relation, and justify your answers. (a,b)R(c,d) if and only if ad=bc. (a,b)R(c,d) if and only if ab=cd. (a,b)R(c,d) if and only if a2+b2=c2+d2. (a,b)R(c,d) if and only if ab=cd.
Part ii
If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence relation.
You just bought a new car from a dealership that sells cars no older than 2015. Consider the relationship to be the “same age.” The relation is “same age” and the set is cars made in 2015, 2016, 2017, 2018, 2019, and 2020. (a) Prove that this relation is an equivalence relation. (b) Describe the partition defined by the equivalence classes
b) If X = \{a,b,c\}X={a,b,c}, give the equivalence classes of the relation.
Answer the following questions on equivalence relation and partial order.
Let S be a nonempty set. Consider the relation ~ on P(S) defined by A ~ B if AUBC = S. Is the relation an equivalence relation? Explain. If yes, identify (no need to justify) the equivalence class of
4. For each relation R below, determine whether it is an equivalence relation on the given set A. Also, find the requested relation class (you can do this even if R isn't an equivalence relation). (a) R= {(1,1), (1, 2), (1, 3), (2, 1), (2, 2), (3, 1), (3, 3), (4, 4)} on the set A = {1,2,3, 4}. Find [1]. (b) R= {(x, y) E R² | x² = y²} on the set A = R. Find [5]. (c) R= {(x,y) E Z² | xy > 0} on the set A = Z. Find [–7].

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