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. uvif 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.

Q: Let R and S be two equivalence relations on a nonempty set, and T their intersection. What's their…

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

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Q: Answer the following question accordingly:

A: Given set is  B={1,2,3,4}

Q: 4. Let A = (a, b,c, d). Suppose R is an equivalence relation on A. Suppose R has three equivalence…

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: 4. Suppose R is a relation on a set A. The complementary relation R is defined by letting a R b if a…

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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…

A: Given below the detailed Solution

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…

A: Given, In the diagram below, there are 9 points and 8 lines. a) Let ⋄ be the relation between the…

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: 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…

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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: 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…

A: Let, R be a relation on L(set of all straight lines in the plane) defined as, xRy iff x and y are…

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: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…

A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric)  So, R∘S=S∘R

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: 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: Given the set S = {1, 2, 3, 4, 5, 6}, define an equivalence relation R. on S where two elements a…

A: Equivalence relations provide a way to classify elements in a set based on shared characteristics.…

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…

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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.

A: We are given  that R is a transitive relation on a set A  Transitive Relation  A relation on a set X…

Q: Which of the following relations E are equivalence relations on the set of integers Z? 3. a. xEy if…

A: A relation E on a set A is called E is reflexive if xEx, ∀x∈A. E is symmetric if xEy, implies that…

Q: If {{1, 3, 5}, {2, 4}} is a partition of the set A = {1, 2, 3, 4, 5}, determine the corresponding…

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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…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by  R=1,1,2,2,3,31,2,2,1 And…

Q: Which of the following relations E are equivalence relations on the set of integers Z? a. xEy if and…

A: Since you have asked question with multiple subparts so as per our guidelines we have to solve the…

Q: Let A Z and let the relation R be defined on A by zRy if and only if 7 divides r-y. a) Show that R…

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Q: Let A = {0,1, 2} × {0, 1, 2, 3}. We define an equivalence relation R on A by saying that (a, b)R(c,…

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Q: Which of the following statements about equivalence relations and partial ordering is/are ALWAYS…

A: The objective is to find which of the following statements about equivalence relations and partial…

Q: Show that if for a relation R we know that R2 C R, then R transitive, and conversely. Ent. There are…

A: To prove whether a relation is transitive or not

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

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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.

Q: 9.13. Let S = {a, b, c}. Then R= {(a,b)} is a relation on S. Which of the properties reflexive,…

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Q: (d) Justifying your answer, find the smallest set of related pairs which must be added to R in order…

A: We know that a relation is called an equivalence relation if it is reflexive, symmetric and…

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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