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

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

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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}?

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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…

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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…