Let G be a connected, undirected graph, and let V be the set of all vertices in G. Define a relation R on V as follows: for any vertices a, b e V, a Rb if there is a path from a tob with an even number of edges. (A path may ise the same edge more than once.) Prove that R is an equivalence relation.

Let G be a connected, undirected graph, and let V be the set of all vertices in G. Define a relation R on V as follows: for any vertices a, b e V, a Rb if there is a path from a tob with an even number of edges. (A path may ise the same edge more than once.) Prove that R is an equivalence relation.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: d. Ris a relation on Z* such that (x, y) e R if and only if there is a positive integer n such that…

Q: 8. Draw the directed graph of the binary relation S on B = { a, b, c, d, e} by S = {(a, b),(b,…

Q: Answer the following question accordingly:

A: Given set is B={1,2,3,4}

Q: Consider the set A ={(1, 3), (2, 4), (-4,-8), (3, 9), (1, 5), (3, 6). Define the relation R on A as…

Q: 13. Let A be the set A = {a, b, c, d}. Consider the relation R on A given by R = {(a, a), (b, b),…

Q: Let A = {2, 3, 4, 5, 6, 7, 8} and define a relation T on A as follows. For every x, y Е А, хТу —31(х…

Q: Let P={{a,b,d},{c,f},{e,g}} be a partition. Write the relation matrix for the implied equivalence…

A: P={{a,b,d},{c,f},{e,g}} . The equivalence relation is as follows:…

Q: 7. Let S = {1,2,3, ...,8}. Let R be the relation on S defined by the following: %3D xRy + xy is a…

Q: Let A = {3,4,5,6} and B = {1,2,3} and define a relation R: A → B as follows: For all (x, y) ∈ A × B,…

Q: Consider the relation on A = {a, b, c, d, e} given by R ={(a,b),(b,c),(b,d),(b,e),(d,e)} .Explain…

A: Given that, A=a,b,c,d,e And R=(a,b),(b,c),(b,d),(b,e),(d,e) The graph for the diagram is,

Q: let A= {a, b, c, d} and R= { (a,a), (a, c),(b,d),(c,a),(c,c),(d,b)} be a relation on A. draw a…

Q: Let A = {6, 8, 13, 15, 9, 16} and define a relation R on A as follows: xRy ⟺ 7 |(x – y) Draw a…

Q: a) Prove that R is an equivalence relation. b) Find [C] for C = {1, 3}. ) How many different…

A: Follow the answer below.

Q: Draw a directed graph for the relation R on the set {0, 1, 2, 3) given by 10. R = {(0, 1), (0, 2),…

Q: State whether or not each of the following is an equivalence relation on the set of vertices of an…

A: An equivalence relation on a set must satisfy three properties: reflexivity, symmetry, and…

Q: Define the relation R on Z as follows: for x, y ∈ Z, xRy if and only y−x ≥ 0. Determine whether R is…

A: partial order means reflexive and antisymmetric.

Q: Create (a) the ordered pairs, (b) graphical representation, and (c) the relation matrix, for the…

A: Given a relation R on the set B as "is factor of" B=1,2,3,6. a The ordered pair is…

Q: Let A = {1, 2, 3, 4, 5} and R be a relation on A such that for all x and y in A, x R y if and only…

A: Given data A={1,2,3,4,5}

Q: Decide whether each of the following relations is an equivalence relation or not. {(a, b) | gpa of…

A: To prove equivalence relation, we have to prove that (i) the relation is reflexive, (ii) the…

Q: lation. Justify why your answers are corre R is a relation on the set of all people sue R is a…

A: EQUIVALENCE RELATION: If a relation R on a set A satisfies the following three criteria, it is…

Q: Let A = {a, b, c, d} and define a relation R on A. The directed graph of the relation is shown here.…

A: A relation R on a set A with elements a, b and c is said to be in a transitive relationship, then if…

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

Q: 6. Let A {1,2, 3, 4, 5} and define a relation on A as follows: For all x, y E A, %3D xRy + x+ y is…

Q: Let C be a relation defined on the set of all geometric polygons, defined by any two polygons in the…

A: The given problem is from relation on sets. Here. Give that C be a relation defined on the set of…

Q: Define a binary relation on the set S=(-3, -2, -1, 1, 2, 3} as follows: R= {(a,b) ab>0} %3D a. Find…

Q: Define a relation on Z by a R b if a² = b². a. Prove that R is an equivalence relation. b. Describe…

Q: Let S = {1,2,3, ...,8}. Let R be the relation on S defined by the following: xRy → xy is a perfect…

Q: Assume a directed acyclic graph is encoded as a relation R. Give a concise proposition stating that…

Q: The relation R=((a,a),(b,b)) on set X=(a,b) is. transitive symmetric reflexive anti-symmetric None…

Q: Côñšidéř thé rélátión S on the set {a, 0, C} given by S = { (a, a), (a, b), (b, a), (b, b), (c, c),…

Q: d b

Q: Let A = {1, 2, 4, 5, 6, 10} and R be a relation on A such that for all x and y in A, x R y if and…

A: Given : A = {1,2,4,5,6,10} xRy if and only if x|y

Q: State whether or not each of the following is an equivalence relation on the set of vertices of an…

A: Equivalence relation is a kind of binary relation that should be reflexive, symmetric and…

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 =…

Question

Let G be a connected, undirected graph, and let V be the set of all vertices
in G. Define a relation R on V as follows: for any vertices a, b e V, a Rb
if there is a path from a to b with an even number of edges. (A path
may 1use the same edge more than once.) Prove that R is an equivalence
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

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 5 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

1. We are given the sets S and V: S = {Anne, Theo} V {studies, runs, meditates} = (a) Find the Cartesian Product of S and V: S x V (b) A relation R. is a subset of the Cartesian product. Create such a subset RC (S X V) with three elements.
3. Let R be a relation defined on the set A = N × N where (a, b) R (c, d) if að = cd. Show R is an equivalence relation. Then, list the elements of the equivalence class [(9, 2)].
For each of the following relations, determine whether the relation is: Transitive. • A partial order. Reflexive. . Anti-reflexive. Symmetric. • Anti-symmetric. A strict order. . An equivalence relation. Justify all your answers. a. R is a relation on the set of all people such that (a,b) = R if and only if a has the same first name or the same last name as b. b. R is a relation on a power set such that (A,B) ER if and only if |A| = |B| (i.e., the cardinality of A is equal to the cardinality of B). c. R is a relation on Z such that (a,b) = R if and only if |a| = |b| +1. d. R is a relation on Z such that (a, b) = R if and only if a+b>1. e. R is a relation on ZxZ such that ((a, b), (c, d)) = R if and only if a+b>c+d.
part b and c
3. Define a relation on the real numbers as follows: x Ry if and only if x² + y² = 1. Determine if R is reflexive, symmetric, and/or transitive. For each property, give sufficient justification for your answer.
Let G be a connected, undirected graph, and let V be the set of all vertices in G. Define a relation R on V as follows: for any vertices a, b = V, a R b if there is a path from a to b with an even number of edges. (A path may use the same edge more than once.) Prove that R is an equivalence relation. Suppose the equivalence relation of Exercise 25 is defined on the vertices of the following graph. What are the equivalence classes? b d f 80 g