Determine whether each relation is an equivalence relation. Justify your answer. If the relation is an equivalence relation, then describe the partition defined by the equivalence classes. Ibegin(enumerate [label=(\alph")] \ item The domain is a group of people. Ferson Zjis related to person yunder relation Mlif zland yhave the same favorite color. You can assume that there is at least one pair in the group, Aand y such that zMy %Enter your answer below this comment line. |item The domain is the set of all integers. ZEif I+is even. Arn integer ais even if z= 24for some integer 411 %Enter vour answer below this comment line.

Determine whether each relation is an equivalence relation. Justify your answer. If the relation is an equivalence relation, then describe the partition defined by the equivalence classes. Ibegin(enumerate [label=(\alph")] \ item The domain is a group of people. Ferson Zjis related to person yunder relation Mlif zland yhave the same favorite color. You can assume that there is at least one pair in the group, Aand y such that zMy %Enter your answer below this comment line. |item The domain is the set of all integers. ZEif I+is even. Arn integer ais even if z= 24for some integer 411 %Enter vour answer below this comment line.

Name: See attached Problem 8Determine whether each relation is an equivalenc
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Description: See attached Problem 8Determine whether each relation is an equivalence relation. Justify your answer. If the relation is an equivalence relation, then describe the partition defined by the equivalence classes.||begin(enumerate}(label=(\ alph")\ item

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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PROBLEM1 Instruction: Prove that the relation is not an equivalence relation. For natural numbers m and n we define m ≤ n if either m = n or m < n.
PROBLEM1 Instruction: Prove that the relation is an equivalence relation. For x, y ∈ R say x is congruent to y modulo Z if x−y is an integer, that is x − y ∈ Z.
subject: discrete structures Which of these relations on the set of all people are equivalence relation?
State the formal definitions that a relation must satisfy in order to count as Reflexive, Symmetric, and Transitive. Then construct three small worlds showing, respectively, that a relation can be Reflexive and not Symmetric, Symmetric and not Transitive, and Reflexive and not Transitive. Finally, for each of the three small worlds that you construct, give an example of a relation in ordinary language that works like the relation in the small world you built. That is, for your first small world, give an ordinary language example of a relation that is Reflexive and not Symmetric.
I know that in order to show an equivalence relation I need to check for (1) reflexivity, (2) Symmetry, and (3) transitivity. I just cant figure out how to prove those things for this problem. (Problem attached.)
Which of the following relations are equivalence relations on the given set S? (provide computations) a) S=R, and a~b <--> a=b or -b b) S=R, and a~b <--> a2+a=b2+b c) S is the set of all people in the world, and a~b means a lives within 100 miles of b. d) S=N, and a~b <--> ab is a square. For those relations that are equivalence relations, describe the equivalence classes.
I have the following question : "Let R be the relation defined on Z by aRb if a + b is even. Show that R is an equivalence relation anddetermine the distinct equivalence classes." I know how to show the equivalence relation, however I have no idea how to approach determining the distinct equivalence classes?
NOTE: This question requires that I find the equivalence classes and not the equivalence relations.
I have the following question and I also have the answer, I just dont understand why and how this question is approached, please elaborate more "Let R = ∅ be the empty relation on a nonempty set A. Which of the properties reflexive, symmetric andtransitive does R possess?"
*NOTE: it says "Let A be a nonempty a set". It does not specify what kind of set, so we may NOT assume it is a relation. Thus A^2 means A x A, NOT composition of relations.*
Problem 7. Which of the following relations ∼ on Z is an equivalence relation? Explain your answer, and for each relation that is an equivalence relation, give the set Z/ ∼ of all equivalence classes. 4.x∼yiffx=yorx=−y.
'' (a,b) ∼ (c,d) ⇔ a+d =b+c '' defined as '∼' relation is that a equivalence relation?