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 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 a g 00 d

Q: Suppose R is a relation defined on the set A = {1,2, 3, 4}. If R is an equivalence relation, what is…

A: Given below the detailed solution

Q: Let A = {3,4,5,6,7,8} and define a relation R on A as follows: For every x, y E A, x R y + 2|(x –…

A: The arrows between two numbers or the number and itself must differ by a multiple of 2 ie, 0,2,4,6,8…

Q: Let R be the relation on the set of integers, s.t., aRb if and only if a=b or a= -b. Is R an…

A:

Q: Determine whether the relation with the directed graph is an equivalence relation:

A:

Q: a)Let R be a relation on the set Z × Z defined by the following rule: For all (a, b),(c, d) ∈ Z × Z,…

A:

Q: Is the following statement true or false? If R is any symmetric relation on a set A, then R is…

A:

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

A:

Q: Answer the following question accordingly:

A: Given set is  B={1,2,3,4}

Q: Given a relation, R on A = {a, b, c, d} on as follows: R = {(а, а), (а, b), (а, d), (b, b), (b, с),…

A: Relation as a Matrix: Let A = {a1,a2,a3,.......am} and B = {b1,b2,b3......bn} are two finite sets,…

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

A:

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

A:

Q: Let F be a nonempty family of transitive relations. Prove that intersection F is a transitive…

A: Given F be a nonempty family of transitive relations. We have to prove that the intersection F is 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…

A:

Q: 3. On the relation R on the set {1, 2, 3, 4, 5} defined by the rule (x, y) e R if x+y<6. a) List the…

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),…

A:

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: r1 1 1 0- 1 0 0 c) The matrix M represents the relations R defined. = 1 1 11 the set A = {1, 2, 3,…

A:

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: 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: The following definitions are used: a relation on a set A is defined to be irreflexive it, and only…

A: The given set is A=01,2,3 The given relation is R=0,0,0,1,0,2,1,2. (i) Using the definitions of…

Q: (b) Let A = {1,2,3}. Consider the relation R on A given by R = {(1,2), (1,3), (2,1), (2,3), (3,2)}…

A:

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

A:

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

A: We have: The relation is: R=a,c,b,a,c,d,d,b

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…

A:

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…

A:

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

A:

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

A:

Q: d b

A:

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: 1. Let A = {2, 3, 4, 5, 6} and B = {10, 11, 12, 13). Define a relation R on (A x B) as follows: a Rb…

A:

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: 1. Let A = {cat,dog,bird,rat} and R be a relation on A defined by {(x,y) : x and y have at least one…

A: We have to solve given problem: