Answered: Let x1, x2, y1, y2 be real numbers. The…

Let x1, x2, y1, y2 be real numbers. The expression (x1, y1) ~ (x2, y2) means that x12 + y12 = x22 + y22. Prove that the relation ~ is an equivalence relation on R2.

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...

See similar textbooks

Related questions

Q: Which of the following relations on the set A = {0, 1, 2, 3} is an equivalence relation? Check all…

A: “Since you have posted a question with multiple sub-parts, we will solve the first three subparts…

Q: disprove: a relation that is symmetric and transitive must TProve also be reflexive. or

Q: Fuppose that the relation R indicates whether c articular student is registered for a particular c…

A: In this question, the concept of Relation is applied. Relation The relation is a relationship…

Q: Let R be a relation on Z defined by (x,y)ER if and only if 5(x-y)=0. Formall state what it means for…

Q: The relation R is defined on R^2 by (x1, y1) ∼ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1) and…

A: a. The relation R is defined on R^2 by (x1, y1) ~ (x2, y2) if and only if x1 − x2 ∈ Z,for (x1, y1)…

Q: Let L be the set of all lines in XY plane and R be the relation in L defined asR = {(L1, L2) : L1 is…

A: Line L is parallel to itself. This means L is parallel to L.So, (L,L)∈RSo, R is reflexive.

Q: Let S be the relation on the set R^2 defined by (x, y) S (w, z) if |x| + |y| = |w| + |z|. Prove or…

A: Definition of an Equivalence Relations: A relation R on a set A is an equivalence relation if…

Q: Let R be a relation on the set A = {x € Z:1 < x < 100} defined as follows: x R y if and only if x-y…

A: Equivalence relation

Q: Is the relation R, defined on the set of real numbers by xRy if 2x + y = 3. antisymmetric?

Q: Define a relation C from R to R as follows: For any(x, y) ∈ R × R, (x, y) ∈ means that x2 + y2 = 4…

A: Topic = function

Q: Let A = Z and B = Z\{0}. Show that the relation R on A x B defined by (a, b)R(c, d) if ad = bc, is…

Q: Let the relation non z defined by x ny if and only if 3/(x+2y) Show that n is an equivalence…

Q: 1) Let S be the equivalence relation on R defined by xSy iff x-y Q. Then a) [√3]-(-√3+r,req) b)…

Q: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: Define a relation on Z by x ~ y if and only if x + 2y is divisible by 3. Show that this relation is…

Q: In the Cartesian plane = {(x, y): x,y E R} consisting of points with rectangular coordinates (x, y),…

Q: Let S = { (x, y) | x, y ∈ R} and relation R on S defined by (x,y) R (p,q) if and only if x-p = y –q…

A: Let S = { (x, y) | x, y ∈ R} and relation R on S defined by (x,y) R (p,q) if and only if x-p = y –q…

Q: Let R be a reflexive and transitive relation on A. Show that RN R¯' is an equivalence relation -1 on…

A: First let, A= set of integers. Then, let, R={(a,b):a,b in A} so,

Q: The domain of relation R is ZxZ. (a, b) is related to (c, d) if a s c and b s d. Which statement…

A: In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and…

Q: Let H be the set of all human beings. Consider the relation R defined on H by xRy if and only if x…

A: R defined on H by xRy if and only if x and y have the same biological mother. reflexivity : x &…

Q: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.

A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…

Q: Let T be the relation defined by, For all a, b e Z, aTb iff a – b is odd. Is T an equivalence…

Q: Check whether R is reflexive, symmetric or transitive and hence determine whether the relation R is…

A: A relation R is said to be reflexive on a set A if for all a∈A, a,a∈R A relation R is said to be…

Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

Q: 1. Consider the relation R1: R - R such that (r1, y1)R1 (X2, y2) if and only if both points are on…

Q: Define a relation C from R to R as follows: For any (x, y) ∈ R x R, (x, y) ∈ C means that x² + y² =…

A: (x,y) ∈ R × R (x,y) ∈ C means x2 + y2 =4

Q: Show that the relation R on Z defined by aRb if and only if 5a−3b is even, for a, b ∈ Z, is an…

A: Let R be a relation from a set A to itself. R is said to be an equivalence relation if 1) R is…

Q: 4. Let R be the relation xRy if 3| (x – y) on Z. Prove that R is an equivalence relation.

Q: A relation R is defined on Z by a R b if |a – b| < 2. Which of the properties reflexive, symmetric…

Q: Check if the following statement is TRUE or FALSE. Let R be the relation on R² defined by…

Q: Determine whether the following relations are equivalence relations. For those relations that are…

Q: Let A- {#, 7, x} and let R be the relation on. A given by R {(7, 7). (#, #)}. Prove or disprove…

A: Transitivity is a property of verbs that relates to whether a verb can take objects and how many…

Q: Which of these relations on {0, 1, 2, 3} are equivalence ?relations

A: Equivalence relations are the relations which are reflexive (xRx), symmetric (xRy implies yRx),…

Q: 1. Consider the relation R1: R2 - R such that (r1, y1)R1 (x2, y2) if and only if both points are on…

Q: Define a relation R on the x-y plane as follows: Given two points (x1 ,y1) and (x2 ,y2) in the x-y…

Q: Let R = {(f, g) | f(1) = g(1)} be a relation defined on the set of func- tions from Z to Z. Is R an…

A: Let R be a relation on a set A. Then R is said to be an equivalence relation if it is reflexive,…

Q: Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) e if and only if (*1, Yı) and (x2, Y2)…

A: The complete solutions are given below

Q: Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive andtransitive but not…

A: Given, the relation R in R defined as R = {(a, b) : a ≤ b}.

Q: Show that the relation R on Z defined by a R b if and only if 5a − 3b is even, for a, b ∈ Z, is an…

A: Given information: The relation R on Z defined by a relation b if and only if 5a − 3b is even, for…

Q: Define a relation on Z x Z, xRy, |x - y| < 1. Is R reflexive, symmetric, and/or transitive? Explain…

Q: Let A = {#, 7, x} and let R be the relation on A given by R = {(7, 7), (#, #)}

A: R is said to be reflexive relation on A if for all x in A, xRx holds. R is said to be symmetric…

Q: Suppose there are two transitive relations R, S Disprove the claim that the composed relation S R…

Q: For universe S = R², define relation ~= as: (a,b) ~= (c,d) < 2a - b = 2c - d Prove that ~= is an…

Q: Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’ and y-y’ are integers. Prove…

Q: 18. Let R be the relation on R defined by xRy → x – y is an integer. Prove that R is an equivalence…

Q: Let R be the relation on N defined by a R b if either a | 2b or b | 2a. Is R an equivalence…

A: image is attached

Q: i Determine whether or not the relation R on the set of all real numbers is reflexiv netric,…

Question

Let x₁, x₂, y₁, y₂ be real numbers. The expression (x_1,y₁) ~ (x₂, y₂) means that x₁² + y₁² = x₂² + y₂². Prove that the relation ~ is an equivalence relation on R².