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.

Q: Draw the directed graph of the relation R = {(1,2), (2,1), (1,1), (2,2), (3,3), (4,1)} X=123 43

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Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

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A: The relation is defined as follows.

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Q: Draw the directed graph of the relation R on A = {1,2,3,4,5,6,7,8} defined as R = {(a,b)|a,b e A and…

A: Here, the relation is: R=(a,b)|a,b∈A and a≡b mod 3 That is: a-b is divisible by 3. So, 7-1=6 is…

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A: Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say  Y~Z if and only…

Q: #14

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Q: efine the relation ~ on the set N of positive integers by a~b if and only if  a = b(10^k) for some…

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Q: that P ◦ Q is an equivalence relation on X iff P ◦ Q = Q ◦ P

A: Concept:

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Q: Consider a relation ~ on the set Z such that x~y if and only if x + y is even. The relation ~ is…

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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

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Q: Let R be a relation over set A. Let A = {1,2, 3, 4, 5, 6} and let R = {(a, b) e A² : |a – b| > 3} a)…

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A: This can be solved as follows:

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Q: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

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Q: 25. Let G be a connected, undirected graph, and let V be the set of all vertices in G. Define a…

A: Equivalence relation: A equivalence relation R on a set A is a relation which is reflexive,…

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

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Q: 3. Determine whether the given pair of graphs is isomorphic. Exhibit an isomorphism or provide a…

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Q: A relation R on a set A is backwards transitive if, and only if, for every r, y, z E A, if rRy and…

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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 =…

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Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

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Q: given the relation R define = (0,0), (0, 1), (0, 3), (1, 1), (1,0), (2, 3), (3, 3) on the set A = 0,…

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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 =…

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Q: Define a relation R on the set J ={ 0, 1, 3, 4, ,5, 6 } as follows for all x, y E J x R y if and…

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Q: Let G be an undirected graph with a loop at every vertex.Show that the relation R on the set of…

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Q: 20 - Determine whether the relation with the directed graph shown is an equivalence relation ? a d

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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…

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Q: 5. Let G=(V, E) be a graph with n > 3 vertices. (a) Define a relation on V by "ry if and only if…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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

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Q: Prove that the given relation is an equivalence relation, and describe the distinct equivalence…

A: Let A is the set of all students at your college. For every x,y∈A,  xSy⇔x is the same age as y. That…