Answered: Let R be a relation defined on Z by xRy…

Let R be a relation defined on Z by xRy if and only if x-y=7k for kez. The equivalence class of [5] is the same as the equivalence class of. with respect to R. O A. [1] O B. [10] C. [12] O D. [4]

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 23E: 23. Let be the equivalence relation on defined by if and only if there exists an element in ...

See similar textbooks

Related questions

Q: Let B = {a, b, c}and S is the relation on B such that S3 (а, a), (а, b), (b, с), (а, с), (с, с).…

A: Given the set B={ a,b,c } and the relation S on B such that S={(a,a),(a,b),(b,c),(a,c),(c,c)}

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. Is R reflexive?…

A: Given: R is a relation defined on P 1, 2, … , 100 by A R B if and only if A ∪ B is even. We have…

Q: A relation < is defined on R? by (x1, x2) < (y1, y2) if and only if x1 < yi and x1 + x2 2 yı + y2.…

Q: Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find…

Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

Q: Define R on Z+ by aRb if = 5 for some integer k. (a) Prove R is an equivalence relation. (b) What is…

A: The solution is as follows:

Q: PROBLEM 30. For (x, y) and (u, v) in R2, define (x, y) ~ (u, v) if x² + y² = u? + v?. Prove that…

Q: b. Define a relation SER XR by S ={ (x,y) E R × R |x – y € Z }. Prove that S is an equivalence…

A: By equivalence relation, we mean that the relation is reflexive, symmetric, and transitive.Here we…

Q: Let E be an cquivalence relation on Z defined by mEn iF and only iF (mtn) is even, where m,n € Z. IS…

Q: Consider the relation R on Z defined by the rule that (a, b) E R if and only if a + 26 is even.…

Q: 10. (A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…

Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…

A: According to the given information, Consider a relation R on R2 as:

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: Q7. Let R be a relation on the set Z defined by the following rule: for all a, b e Z, a Rb if and…

Q: Consider the binary relation R defined on (N* x N*) by: V (X1. Yı). (x2. Y2) E N* x N*, (x1. Yı) R…

A: Order relation (definition)-: A relation R is called an order relation if it is reflexive,…

Q: Let S be the following relation on C\{0}: S = {(x, y) = (C\{0})² : y/x is real}. Prove that S is an…

Q: 6. Consider the relation R on Z defined by xRy iff x – y= 4n for some n Ɛ Z. (a) Show that R is an…

Q: A relation R defined on the given set A will satisfy

Q: Let X = R? – (0,0) and - be defined over X by setting X~y + y = tx for some non-zero t e R. (b)…

A: Sol

Q: 10. (1) A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe 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: 4. For r, y E R, let z y if and only if (x- y) E Q. Show that defined as such is an equivale…

A: ~ is defined as : For all, x, y in R, x~y if and only if (x-y) belongs to Q Reflexivity : x~x for…

Q: on K³defined by: (a, b, c) ~ (d, e, f) if and only if ak e K – {0} such that (a, b, c) = k(c, e, d)…

Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…

A: Consider the given information.

Q: Let R be the relation on N defined by x R y if x and y share a common factor other than 1. Determine…

A: The relation R is defined as xRy on natural numbers Nsuch that x and y share a common factor other…

Q: Problem 17. (a) Prove that <r is a reflexive, transitive relation on P(E*). (b) Prove that =r is an…

Q: Let S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an…

Q: Let R be a relation from X to Y and S be its reverse relation from Y to X. Prove or disprove that R…

A: According to the given information it is required to prove/disprove that R is injective and well…

Q: Consider the relation on Z×(Z\ {0}) defined by (m,n)R(m’,n’) provided that mn’ = m’n. Prove that…

Q: 3. I Consider the relation R= {(r, y) | x+y is even} on the set Z of integers. Show %3D that R is an…

Q: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

Q: Let R be the relation defined on P({1,., 100}) by ARB if and only if |AU B| is even. s R reflexive?…

A: We will check one by one what properties R satisfies and which not.

Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in R…

Q: . Let R be the relation on N defined by xR y if x and y share a common factor other than 1.…

Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

Q: 2. Let H ≤ G and define = on G by a = b iff a¯¹b € H. Show that =µ is an equivalence relation.

Q: Let A = Z. Define R to be the relation where a~b if ab > 0. Then R is equivalence relation.

A: We have given the set A = { Z : the set of integers } R= { (a,b) : a,b belongs to Z and a.b >0}…

Q: 14. For a,b E Z, define a ~ b if and only if 4a + b is a multiple of 5. Show that defines an…

Q: Given a relation R on Z defined by mRn 3|(m – n) for all m, n E Z. Find the equivalent class…

Q: Let R be the relation defined on P({1,.…, 100}) by ... ARB if and only if |AUB| is even. Is R…

A: Relation: - A subset of the productA× B could be a relation R from a non-empty set A to a non-empty…

Q: In this exercise, you want to show that a relation RC A? is transitive + R" C R for all n EN=…

Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…

A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…

Q: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…

A: A relation to be an equivalence relation must satisfy the following three properties: 1. Reflexive…

Q: Let S be the following relation on C: S = {(x, y) = C²:y-x is real}. Prove that S is an equivalence…

Q: Define a relation on Z by setting xRy if xy is even. a. Give a counterexample to show that R is not…

Q: For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence…

Question

Let R be a relation defined on Z by xRy if and only if x-y=7k for kez. The equivalence class of [5] is the same as the
equivalence class of
O A. [1]
O B. [10]
O C. [12]
_with respect to R.
O D. [4]

Expert Solution

This question has been solved!

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

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Step by step

Solved in 2 steps

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

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.
For each of the following parts, give an example of a mapping from E to E that satisfies the given conditions. a. one-to-one and onto b. one-to-one and not onto c. onto and not one-to-one d. not one-to-one and not onto
For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .
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 .