State whether or not each of the following is an equivalence relation on the set of vertices of an arbitrary graph. In each case give a proof or a counterexample. i. uv if there is an edge between u and v. ii. uv if there is no edge between u and v. iii. uv if there is a path between u and v.

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Q: (24 pts) Either show the following relations are equivalence relations, or show which qualities of…

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Q: Proposition 2.12. Suppose- that R is an equivalence relation on a set A, and let a, b e A. Then the…

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Q: Give an example of two equivalence relations R and S on

A: Consider that R and S are equivalence relations on A=1, 2, 3.

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

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Q: 1. In the diagram below, there are 9 points and 8 lines: (a) Let be the relation between the points…

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Q: 8. Define a relation R on the set J = {0, 1,3,4,5,6} as follows: For all x, y EJ, x Ry ⇒ 4x² + y².…

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Q: Let A = {2,4,6}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule     (X,Y) ∈ R ⇔…

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

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Q: S is an equivalence relation. 2. Define the relation R on Z as follows: for x, y E Z, rRy if and…

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Q: Let I be the set of all straight lines in the plane. Define and equivalence relation R by x R y iff…

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Q: . Let A = {1,2,3,4}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule     (X,Y) ∈ R…

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

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Q: 3. Given a relation R on a set A, prove that if R is transitive, then so is R-1.

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Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

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Q: 1. Let A = {1,3,5}, let S = P(A) and let R ⊆ S x S be a relation defined by the rule     (X,Y) ∈ R…

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Q: d b

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Q: (a) Let A = {a} and B = {1}. List all relations from A to B. (*Hint*) {a} and B = {1,2}. List all…

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Q: Draw the Hasse diagram for the relation C on the set P({1,2,3}).

A: According to the given information, it is required to draw the hasse diagram for the given relation.

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Q: 4. Prove that the next relation is equivalence relation on the set of all people: {(x, y)|x and y…

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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…