PROBLEM 3A car dealership sells cars that were made in 2015 through 2020. Let the cars for sale be the domainof a relation R where two cars are related if they were made in the same year.Prove that this relation is an equivalence relation.Describe the partition defined by the equivalence classes.

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Answered: PROBLEM 3 A car dealership sells cars…

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015

1st Edition

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:HOUGHTON MIFFLIN HARCOURT

Chapter4: Writing Linear Equations

Section: Chapter Questions

Problem 7CA

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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.
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.
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.
Problem 3The power set of any set A is denoted by P (A) and is defined as the set of all subsets ofA, including the empty set and the set A itself. Think of ⊆ as defining a relation on P (A)where P, Q ∈ P (A) are related if and only if P ⊆ Q. Is this relation reflexive, symmetric,and/or transitive?
Problem 2Let A be a set, and let R be a relation on A. Suppose that R is symmetric and transitive.Find the flaw in the following alleged proof that this relation is necessarily reflexive. “Letx ∈A. Choose y ∈A such that xRy. By symmetry know that yRx, and then by transitivitywe see that xRx. Hence R is reflexive.”
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.
NOTE: This question requires that I find the equivalence classes and not the equivalence relations.
Problem 4Let A be a set, and let R be a relation on A.1. Suppose R is reflexive. Prove that ∪x∈A[x] = A.2. Suppose R is symmetric. Prove that x ∈[y] if and only if y ∈[x], for all x, y ∈A.3. Suppose R is transitive. Prove that if xRy, then [y] ⊆[x] for all x, y ∈A.
'' (a,b) ∼ (c,d) ⇔ a+d =b+c '' defined as '∼' relation is that a equivalence relation?
Problem 6. Consider the relation R = {(1,3), (3,1), (2, 3), (3, 3), (3, 4), (4, 4), (4, 1)} on set {1, 2, 3, 4}. a) Find the reflexive closure of R. b) Find the symmetric closure of R. c) Find the transitive closure of R.
b) If X = \{a,b,c\}X={a,b,c}, give the equivalence classes of the relation.
Suppose A and B are two finite sets such that |B| = |A| - 1 and the number of relations on A is 512. Determine: |B|: Number of relations from B to A:

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PROBLEM 3 A car dealership sells cars that were made in 2015 through 2020. Let the cars for sale be the domain of a relation R where two cars are related if they were made in the same year. Prove that this relation is an equivalence relation.