Let G = (V, E) be a graph. Define a relation R on V ly: for all v, tz € V,we have v, Rez if and only if deg (v) = deg(vs).(a) Prove R is an equivalence relation(b) Take G = (V, E) to be the wheel graph on 5+1 vertices, W. What partition of Vis determined by the equivalence classes of the equivalence relation above?

Answered: Image /qna-images/answer/b13adba0-d28e-4691-95b6-98a1c94f2f12.jpg

Answered: Let G = (V, E) be a graph. Define a…

Let G = (V, E) be a graph. Define a relation R on V ly: for all v, y EV, we have v, Rvy if and only if deg (v) = deg(v). %3D (a) Prove Ris an equivalence relation

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 12E: Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and...

See similar textbooks

Related questions

Q: Let A C B and let S be an equivalence relation on B. Prove that R = {(x,y) E A × A : (x, y) E S} %3D…

Q: Consider the following relation R on R: (x, y) ER if and only if y E Q. What are the properties of…

Q: . Define a relation R from R to R as follows: For every (x, y) ERX R, (x, y) E R means that y = r.…

Q: Let A = Z. Define R to be the relation where a-b if ab > 0. Then R is equivalence %3D relation.

A: A relation R on a non empty set A is said to be equivalence relation if R is reflexive, symmetric…

Q: Let f : A → B be an injective function, a be an order relation on B and 3-1 = {(x, y) E A x A :…

A: An order relation means the criterion of comparison between objects, that satisfies the properties…

Q: B and let R = {(x, y) E A x A | g(x) = g(v)}. Suppose g: A Show that R is an equivalence relation on…

A: Hello. Since you have posted multiple questions and not specified which question needs to be solved,…

Q: (a) Prove the following relation: B(x + 1, y): B(x,y) x + y %3D

Q: Let R be the relation defined by x R y =x<y° Determine if R is an order relation on N? on Z? Justify…

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

Q: A function F from a set A to a set B is a relation with domain A and co-domain B that satisfies the…

A: Relation: Let A and B be two sets then relation between A nd B means the collection of the…

Q: Let R be the relation defined by x R y x² < y² Determine if R is an order relation on N? on Z?…

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

Q: Let S {(a,b): a,be R, a 0}. Show that the relation R defined on S by (a, b) R(c,d) if = ad be is an…

Q: Give the relation R induced by the partition above on A.

Q: For each n E N, we write S" := {(x1, x2, ..., In+1) E R"+1 : E? = 1}. (1) Let f : S2 → Rª be the…

A: Given: Sn:=x1,x2,...,xn+1∈ℝn+1:∑i=1n+1xi2=1 for each n∈ℕ f:S2→ℝ4 is the function defined by…

Q: 1. Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

Q: Let g : R x N R be defined by g(x, y) 3 xy 1. Find the image of R x Nunder g, that is, g(R x N).…

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2,y2) if x1y2 = X2y1. Also define g: X [1,2] by the…

Q: Let R be a relation from A={−1,0,1,2} to B={a,n,i} such that (x,y) provided that x is non-negative…

Q: Consider the relation R on R defined by (x, y) E R if and only if |æ – y| = ||x| – ly||- This…

Q: Let ≥ be a relation defined on sets A and B by A ≥ B iff there exists a surjective function f : A →…

Q: Given n e N and the function f : R" → R", we can define, for each nonnegative integer k, the kth…

A: (f) Given : f : ℝn → ℝn is linear function, where n∈ℕ. To show : We have to check whether the…

Q: Let R be a relation on Z. Then Ris an equivalence relation if it is defined by xRy if and only if…

A: R is an equivalent relation iff R is reflexive. R is symmetric. R is transitive.

Q: Let (X, T) be a topological Space and ACX- Develop the relation between Fr ( Fr(A)) and Fr(A).

Q: Verify that the relation "... is tangent to... at a" is an equivalence relation on the functions…

A: An equivalence relation on a set S is a relation ~ on S such that: 1) x~x for all x∈S then the…

Q: (b) Every continuous map S? →T² is homotopic to the constant map.

Q: 5. Let R be a relation on R2 defined by (w1,y1)R(x2y2) if and only if yı +x1 = y2 + x2. (a) Is R…

A: Given relation is

Q: A relation R on a set A is called irreflexive iff (Vx E A)((x, x) ¢ R). It is called asymmetric iff…

Q: Consider X = [1,2]? with the relation (x1, y1)~(x2, Y2) if x1Y2 = x2y1. Also define 3x g: X → [1,2]…

A: We shall solve first three subparts only as per company's guidelines. For others please post again…

Q: Let f: X→Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Let E be…

Q: Part II: Prove/Disprove the following 2. Let R be the relation on Z, defined as follows: Vm, n e Z,…

Q: Let S = {R| R is an equivalence relation on z†}. Prove that S 3 P(Z+). (Hint: Recall that for an…

A: Let A, B and C are the sets. For reflexive , Let a∈A , A⊂-A It is reflexive. Let A⊂-B and B⊂-C ,…

Q: Let f : X → Y be a function. Define a relation R in X given byR = {(a, b): f(a) = f(b)}. Examine…

A: Given:

Q: Let A = {u, v, w}, B = {x, y, z}. Determine whether the relation R = {(u, x), (v, z), (w, y)} from A…

A: In R from A to B is an injection function as well as a surjection function.

Q: Let g be a vertical stretch by a factor of 5, followed by a translation 4 units up of the graph of…

Q: Define a relation T on R by xT y if and only if (sin^2) x + (cos^2) y = 1. (a) Prove that T is an…

Q: 8. Let A = Z* U {0}, define relation R on A × Z* by (k,l)R(m,n) → kn = lm. (a). Prove that R is an…

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: RUR and RNR symmetric relations on X. are Show that R=Z xZ 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: Let f: X - Y and g : Y - X be maps between sets. Suppose that g•f is equal to the identity map idy :…

Q: - Use the connectivity relation Mg- MR V M V M to find the transitive 2. closure of the following…

A: To find- Use the connectivity relation MR* = MR ∨ MR2 ∨ MR3 to find the transitive closure of the…

Q: In this exercise, you want to show that a relation RC A? is transitive + R"C R for all n E N = {1,2,…

Q: Determine whether the following relation is a function from R into R. A Iso deter- mine whether that…

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 G = (V, E) be a graph. Define a relation R on V ly: for all v, tz € V,
we have v, Rez if and only if deg (v) = deg(vs).
(a) Prove R is an equivalence relation
(b) Take G = (V, E) to be the wheel graph on 5+1 vertices, W. What partition of V
is determined by the equivalence classes of the equivalence relation above?

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 4 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, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Does the relation is a brother of have a reflexive property consider one male? A symmetric property consider two males? A transitive property consider three males?
Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.
Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x + 1 }(i) Depict this relation using an arrow diagram.(ii) Write down the domain, codomain and range of R.
= {(1,2), (2,1), (1,1), (2,2), (3,3), (4,1)} Draw the directed graph of the relation R X= {1,2,3,4}.
Find the smallest relation containing the relation {(1, 2), (2, 3), (2, 4), (3, 1)} on theset {1, 2, 3, 4} that is reflexive, symmetric and transitive.
Consider the plane (P) related to x-axis and y-axis system. Let R be the binary relation on (P) defined by: М (х, у) R M'(x', у') —x-у%3х' - у' |(a) Prove that R is an equivalence relation on the set of points of (P). (b) Interprete geometrically the equivalence class of the point A (1, 2). (c) Describe geometrically the set (P)/R.
11. Prove that the relation on L(V, V) given by similarity is an equivalence relation.
2) Let Q = {0,1,2,3} and define relation R on Q as follows: R = {(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0), (3,3)} Determine whether the R relation is reflexive, symmetric or transitive.
Determine if the relation R = {(1,3), (1,4), (2,3), (2,4), (3,4)} is reflexive, symmetric, antisymmetric, or transitive.
State whether the statement is true or false :'' All relations are mapping but the converse is not true ''.
(c). Given the relation R = {(1,2),(2,3)} on the set A = {1,2,3} . What is the minimum number of ordered pairs which when added to R make it an equivalence relation?Discuss your reasoning.
In each case determine if the given relation R on A is an equivalence relation.Justify your answer.(g) A = R2 and (a, b)R(c, d) if a = c.(h) A = R2 and (a, b)R(c, d) if a^2+b^2=c^2+d^2.