6. Let (a, b), (c,d) E R2. Define a relation ~on R2 by [ (a, b)~ (c,d) if |ab|=|cd| Show that is an equivalence relation. Find the equivalence class of (1,0), (0,0), (0,3).

Q: Let A = {1,2,3,4} and B = {5, 6, 7}. Let f be the relation f = {(1,5). (2,5), (3,6), (?, ?)} where…

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

A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…

Q: Prove that the relation of congruence is an equivalence relation.

A: A relation is said to be an equivalence relation if and only if the relation is reflexive, symmetric…

Q: 4. Let R be a relation defined on the set of integers by a R b if a +b = 0 or a – b = 0. Determine…

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Q: Which of these relations on the set {0, 1, 2, 3} are equivalence relations? Determine the…

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Q: 7. Let (a, b), (c,d) E R2. Define a relation ~on R² by (a, b) (c,d) if 2a - b = 2c-d. (a) Show that…

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Q: 2. Let R = {(x, y) = R x R = |x] + [y] =4}. Then R is a relation on R. Is R an equivalence relation…

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Q: 4. For (1, yı) and (r2,y2) in R² we set (11. yı) ~ (12. Y2) →{+ =3+ yi. (a) Show that ~ defines an…

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Q: ' (a,b) ∼ (c,d) ⇔ a+d =b+c '' defined as '∼' relation is that a equivalence relation?

A: It is equivalence relation if (a, b) belongs to ZxZ

Q: Show that the relation R on ZxZ defined by (a, b) R (c, d) if and only if a + d = b + c is an…

A: To show that the relation R on Z×Z defined by(a, b) R (c, d) if and only if a+ d = b+ c is an…

Q: 2 Explain why the relation on N defined as is not an equivalence relation. {(x,y): +y is a multiple…

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Q: Give a specific reason why the following set R does not define an equivalence relation on the set…

A: We will find out the required value.

Q: Let S be the relation on the set R^2 defined by (x, y) S (w, z) if |x| + |y| = |w| + |z|. Prove or…

A: Definition of an Equivalence Relations: A relation R on a set A is an equivalence relation if…

Q: Let a relation ∼ on R2 = {(a, b): a, b ∈ R} be defined as: (a, b) ∼ (c, d) if and only if | a | + |…

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Q: 5. Let a and b be real numbers, not both zero. For (71, y1) and (12, 42) in R², set (11, yı) ~ (12,…

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Q: on P(X), where X = Y, Z E P(X), we say Y - Z if and only if Y has the same number of ele- Define a…

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: Let R=}(x,y)|2 divides 8x+by} be a relation the Set H= {1, 2,3.4,5, 6. 10 Determine whether the…

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Q: Let A = Z and B = Z\{0}. Show that the relation R on A x B defined by (a, b)R(c, d) if ad = bc, is…

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Q: 22) (a) Show that the relation {(0,0), (1,1),(1,2), (2,1), (2,2), (3,3)} is an equivalence relation…

A: Given 0,0,1,1,1,2,2,1,2,2,3,2

Q: Define a relation on Z by x ~ y if and only if x + 2y is divisible by 3. Show that this relation is…

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Q: Show that the following is an equivalence relation: R = {(1, 1), (1, 3), (1, 5), (2, 2), (2, 4), (3,…

A: Equivalence relations are a fundamental concept in mathematics that allow us to group elements of a…

Q: (3) Let S be the equivalence relation on (0, 1, 2, 3} x {0, 1,2} defined by (a, b)S(c, d) if and…

A: From the given information. “S” is an equivalence relation on {0,1,2,3}X{0,1,2}. The equivalence…

Q: 3. Let S = {1, 2, 3, ...}. Let R be a relation on he set S x S defined by (a, b) R (c,d) ⇒…

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Q: (3) Let S be the equivalence relation on {0, 1, 2, 3} x {0, 1,2} defined by (a, b)S(c, d) if and…

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Q: 4. For each relation R below, determine whether it is an equivalence relation on the given set A.…

A: Definition (Equivalence relation): A relation R is said to be an equivalence relation on a set A if…

Q: 1. Suppose A = {1,2, 3, 4} and the relation R on A is R = {(1,1), (2, 2), (3,3), (3, 4), (4, 4)} (a)…

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Q: 1) Let R be the relation on Z+ such that (p, q) R(r, s) if and only if p- s = q ·r. That is, the…

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Q: (1) Show v is an equivalence relation on S. (2) Describe the equivalence classes of . (3) Describe…

A: Given: Define ~ on S=a, b: a, b∈ℤ, b≠0 by a, b~c, d if and only if ad-bc=0. To show: 1) ~ is an…

Q: 3. For universe S = R² define relation ~ as: (a, b) ~ (c,d) ⇒ 2a-b2c - d. Prove that is an…

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Q: 8. (6 points) Define a relation R on Z as follows: For all integers m and n, m R n ⇒ (m − n)is a…

A: The objective of the question is to understand the properties of the relation R defined on the set…

Q: Let be the relation defined on R by the condition that x - y if x³-y³ = 13(x - y). For example, 1-3,…

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Q: Lex 172,3,4). Consider the relation R on X defined as follows. R = ((1.1), 13), (2,2), (2.4), (3,1),…

A: I have solved both subpart using basic definition of equivalence relation and equivalence class.

Q: 13. Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of…

A: The relations on 0, 1, 2, 3 are given as: (a) 0, 0, 0, 2, 2, 0, 2, 2, 2, 3, 3, 2, 3, 3 (b) 0, 0, 0,…

Q: 6. Consider the set R2 and relation~: (1, 1)~ (2, 2) if and only if x2 = ₁ + m and y2 = ₁ +n for…

A: We shall solve only the first and second part of the question. If you want to solve the remaining…

Q: 3. I Consider the relation R= {(r, y) | x+y is even} on the set Z of integers. Show %3D that R is an…

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