Consider the collection of sets {An: n E R}, where for each n E R An := {(x, y) E R²: 2x + n}. y = (a) Show that this collection of sets partitions R², and sketch several of the sets An. (Note: The index set here is R, meaning that n is a real number!). (b) Find an equivalence relation {An: n E R}. ~ on R? whose equivalence classes are precisely the sets

Q: Which of the following sets is not closed for the operation °, which is defined by the rule a°b = a…

A: The objective of this question is to determine which set of integers is not closed under the…

Q: - Recall that for x, y € R, x - y = R is the absolute value of x-y. That is, if x-y 20 if x-y<0. x-y…

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Q: Let A be a set, and denote by the set of orderings of A. That is, A = {(A, <)< is an order relation…

A: The main objective is to show ~ is equivalence relation.

Q: Let ∼be the relation on Z defined as follows: For every m, n ∈Z, m ∼n if, and only if, 5 | (m2 −n2…

A: (a) For every m ∈ Z, 5 | (m2 −m2) = 0      Hence, m ~ m for all m ∈ Z      Hence, ~ is reflexive.

Q: Q4) Find the following: 1- Find the relation Z from C to X where C = {1, 2, 3, 5} and X = {4, 6, 9}.…

A: Here, C=1,2,3,5X=4,6,9

Q: Q4) Let S be the set of registration numbers of students enrolled in a BSCS program in a university…

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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: Let S be the set of all ordered pairs of non-negative integers. Let R be a relation defined on A…

A: We have to find the equivalence of the given function and we are also given with the set in which…

Q: We have defined the relations ≡6 and ≡10 to be certain subsets of Z^2 . Their intersection R =…

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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: Suppose that f : A → B is a function, viewed as a relation (set of ordered pairs). Let f* denote the…

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Q: How can I prove that the relationship r^2 is a subset of r using the definition of composition?

A: The objective of this question is to prove that if r is a transitive relation on a set A, then r^2…

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: 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: A relation ~ is defined on the set of all pairs of integers by the formula (m,n) ~ (u,v) m - An =…

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Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…

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Q: 9. The relation ~ defined on the set of natural numbers by a ~ b means 'a – b is even is : A. an…

A: Equivalence relation : A relation which is reflexive, symmetric and transitive is an equivalence…

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: ) For arbitrary m, n € N, let (m, n) denote the greatest common divisor of m and n. or example,…

A: A relation ~ on any set R is said to be equivalence iff i) R is reflexive i.e a~a for all a∈R ii)R…

Q: What are the equivalence classes of the equivalence relations on {0, 1, 2, 3}?

A: Answer:-  Find the equivalence classes of the equivalence relations on A = {0, 1, 2, 3}?

Q: Let R be a relation on A. Then for all n ≥ 0, prove R^n is a set. (Discrete Mathematics, powers of a…

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Q: In each of the following cases of a set X and relation determine whether ~ is reflexive, symmetric,…

A: “Since you have posted multiple questions, we will provide the solution only to the first question…

Q: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

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Q: 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…

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Q: Let R be the relation defined on Z by aRb if a + b is even. I have the following equivalence classes…

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Q: A relation R is defined on the set of natural numbers N={1,2,3...}  as follows: aRb means "b=a+6/a;…

A: Domain means all the values that can be taken so that the aRb is defined.

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: Which of these relations on the set of all functions from ℤ to ℤ are equivalence relations?…

A: (a) R = {(f, g) | f (1) = g(1)}        Now, as f(1) = f(1), thus (f, f) ∈ R.              Hence, R…

Q: (Chapter 11.4, #3) The relation relation. Sketch it, by drawing Z as a subset of the line, then…

A: The objective is to show that x≡ymod4 is an equivalence relation. To show the relation 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: Exercises 1-10, determine whether the given relation is an uivalence relation on (1, 2, 3, 4, 5). If…

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Q: I.A) Antisymmetric Relation * A relation R on a set A is called antisymmetric if vavb((a,b) ER^…

A: I. Here, we have to select the best answer for each item. I. A) Antisymmetric Relation. I. B)…

Q: 27. If an equivalence relation has only one equivalence class, what must the relation look like?

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Q: Which of the following sets is not closed for the operation °, which is defined by the rule a°b = a…

A: Option (a) is not closed under the operation °.Option (b), (c), and (d) are closed under the…

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

A: