1. Consider the set Z in which the relation R is defined by aRb if and onlyif a + 3b is divisible by 4, for a, b E Z. Show that R is a reflexiverelation on set Z.2. A relation p is defined on the set of all real numbers R by xpy if andonly if |x- yl < y, for x, y E R. Show that the relation p is notreflexive relation.3. Let m be given fixed positive integer. Let R = {(a,b): a, b € Z} and(ab) is divisible by m. Show that R is symmetric relation.

Since you have asked multiple question, we will solve the first question for you. If youwant any…

Answered: Consider the set Z in which the…

Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible by 4, for a, b EZ. Show that R is a reflexive relation on set Z.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 2E: 2. In each of the following parts, a relation is defined on the set of all integers. Determine in...

See similar textbooks

Related questions

Q: 1. Given a relation R on a set A, prove that if R is transitive, then so is R-

A: Transitive means aRb and bRc then aRc

Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: Consider the relation R={(a, b),(a, c),(c, b),(b, c)} on the set A={a, b, c}. Is R reflexive? Is R…

Q: Given a non empty set X, consider P(X) which is the set of all subsets of X. Define the…

A: The relation is defined as follows.

Q: In each item below, verify that ~ is an equivalence relation on the given set. Let S = Z × N. Define…

A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…

Q: 1. Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible…

A: According to question given that Set Z in which the relation R is defined by aRbIf and only if a+3b…

Q: A relationR on a set A is called equivalence if R is:

A: Equivalence Relation: A relation R on a set A is said to be an equivalence relation if and only if…

Q: Show that “is unitarily equivalent to” is an equivalence relation on Mn×n(C).

Q: Let Z be the set of integers. Consider R = {(r, y) e Z x Z: 7 divides x-y} CZ x Z. Show that R is an…

Q: Let S be the set of integers. If a, b E S, define aRb if a + b is even.Prove that R is an…

Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

Q: Let R be a transitive relation on set A. Define Q to be {(r Prove or disprove that (RUR®)n (QU Qº)…

Q: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

Q: Define the relation on the set Z of integers by if and only if a ~b if and only if a = b + 5k for…

Q: Use a membership table to prove the following relation: * (An B) U (A n Bº) = A

A: We have to use membership table.

Q: Show that the relation R defined in the set A of all polygons as R = {(P1, P2) :P1 and P2 have same…

A: Given, relation R defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have same number…

Q: Consider the following relation R on the set Give an example of a pair that is not in R, but in the…

Q: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

Q: 1. Let & be the relation of (a,b) la #b] on the set of integers. What is the reflexive closure of R?

A: As per our company guideline we are supposed to answer only one question with its 3 subparts. Here…

Q: A particular relation * is defined on the set Z-{0} by x * y if and only if every prime factor of x…

A: Given that a particular relation is defined on the set ℤ-0 by…

Q: Let P be a partition of the nonempty set A. For x, y E A, define xQy if and only if there exists CEP…

Q: Suppose A,S CU, (<) e LOrd(A), and B (a1,b1), (a2, b2) E (A × B): (S). If we define a relation (3)…

A: Suppose x=a,b∈A×B is an upper bound of the two point set. We have show that…

Q: In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z x N.…

Q: 3. Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: We are given that R is a transitive relation on a set A Transitive Relation A relation on a set X…

Q: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

Q: et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈…

A: This is the problem of Relation.

Q: Let the relation R on the set Z be given by the condition: (x, y) E Rif and only if x4 = y4. Answer…

Q: Let N be the set of all natural numbers and let R be a relation in N, defined by R = {(a, b)} : a is…

Q: Consider the subset relation C on the power set of {a,b,c}, that is P{a,b,c}. (a) Construct the…

A: Let S be a partially ordered set. Use the following definitions, to obtain the solution. Let a,b∈S…

Q: A relation R, is defined on Z, the set of integers by aRb + a - b is even. Show that R is…

Q: Let A={(m,n)∣m,n∈N and gcd(m,n)=1} and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc…

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

Q: Let X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…

Q: Use a particular counterexample to explain why R fails to be an equivalence relation is the…

A: We take counter example 9R10,10R11⇒9R 11

Q: Determine, with justification, whether the following relations are equivalence relations. If so,…

Q: . Draw the Hasse diagram of the relation R on A, where A = (1, 2, 3, 4}, and R = {(1, 1), (1, 2),…

Q: Consider the equivalence relation E on N defined by Enm iff n-m is divisible by 4. Which statement…

A: An equivalence relation E is defined on the set of natural numbers N as Enm denotes that n is…

Q: Let X be a set. Let P be a set of subsets of X such that: • ØEP; • the union of all sets A E P is X.…

A: Solution:-

Q: Let A be a set and R be a reflexive relation on A. Prove that R is an equivalence relation if and…

Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

Q: For (a, b),(c,d) EN×N define the relation (a, b) ~ (b,c) if and only if ab = cd. Is this relation…

A: Explanation of the answer is as follows

Q: Let A={(m,n)∣m,n∈N2 and gcd(m,n)=1}, and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc.…

Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

Q: Determine which of the folowing relations R on Z are oquivaknee relatinns. If R is not an equivakner…

Q: Let G={(m,n)∣m,n∈N x N and gcd(m,n)=1}, and define the relation q on G according to (a,b) q…

A: A totally ordered set is a set G together with a relation ≤ that satisfies the following conditions.…

Q: Let R be an equivalence relation on A. Find the maximum number of classes R can make if JA| = 80 and…

Q: Consider the relation R on the set Z defined by a R b if and only if 3|(a+2b) for a, b ∈ Z. Show…

Q: Determine whether the relation ? on the set of all people is reflexive, symmetric, antisymmetric,…

A: We have to determine whether the relation R on the set defined in the parts (a) and (b), are…

Question

1. Consider the set Z in which the relation R is defined by aRb if and only
if a + 3b is divisible by 4, for a, b E Z. Show that R is a reflexive
relation on set Z.
2. A relation p is defined on the set of all real numbers R by xpy if and
only if |x- yl < y, for x, y E R. Show that the relation p is not
reflexive relation.
3. Let m be given fixed positive integer. Let R = {(a,b): a, b € Z} and
(ab) is divisible by m. Show that R is symmetric relation.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step 3

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 3 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Relations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

Similar questions

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 .
Let (A) be the power set of the nonempty set A, and let C denote a fixed subset of A. Define R on (A) by xRy if and only if xC=yC. Prove that R is an equivalence relation on (A).
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.
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 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.
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.