Let R = {(a, b) | a = b (mod 3)} be an equivalence relation on the set of integers Z. What are the three equivalence classes of R? Provide five integers that are members of each equivalence class.

Q: Let J={1,2,3,…}, the set of counting numbers and let R be the set of real numbers. Is R-J countable?…

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Q: Define the following infinite collection of subsets of the positive integers: A1 = {x|x E Z* and 0 <…

A: A relation R is said to be an equivalence relation, if it is  Reflexive : For every a∈R→(a,a)∈R…

Q: Let A = {74, 77, 83, 89, 92, 112, 119, 134, 148} and R be an equivalence relation defined on A where…

A: Given A={74,77,83,89,92,112,119,134,148} For a∈A  class of a=[a]={x∈A : xRa }={ x∈A : x≡a mod 5 } ={…

Q: Let A be the set of all citizens of New York City. For a, b E A, define a ~ -b if and only if ●…

A: To show that ~ defines an equivalence relation on the set A (the set of all citizens of New York…

Q: The congruence modulo 5 relation is an equivalence relation on the set {0, 1, 2, 3, 4, 5, 6, 7,8,9,…

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Q: And what would be the 15 equivalence relations? And what does "up to isomorphism" mean?

A: Ans-In this question we have to determine   all possible equivalence relations on the set…

Q: ) Let Z denote the set of all integers. Determine whether each of the relations on Z given below are…

A: Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: reflexive, symmetric, antisymmetric, transitive they have. Write which of these is an equivalence…

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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 be the set of all permutations of the 26 letters in the English alphabet. The relation R on A…

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Q: count the number of possible equivalence relations on a set of size n by providing a formula in…

A: We use the idea of partitioning to get the total number of equivalence relations.

Q: Let R ⊆ ℝ×ℝ with R={(x,y)|⌈x⌉=⌈y⌉}, that is, x and y round up to the same number. Characterize R in…

A: Given problem is :

Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

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Q: Prove that the relation ∼ between sets is symmetric and transitive.

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Q: Give an example of an equivalence relation R on the set Q such that there are exactly three distinct…

A: We take the help of the signum function

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: 2. Let R be a relation defined on the set of integers Z where a Rb if and only if a? = b° (mod 5).…

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Q: 2) Let m +0 be an integer. Show that Rm := { (a, b) E Z × Z|m divides a – b} is an equivalence…

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Q: Let T: IR2 > R2 reflection (or orthogonal symmetry) with respect to the line 2x - y = 0. We call…

A: Given that T:ℝ2→ℝ2 is a reflection with respect to the line 2x-y=0. Also, given a=2-1, b=12. a)

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

A: A relation R on a set X is a subset of X×X. That is R⊆X×X. (b). Let ≡6 and ≡10 are two subset on the…

Q: Consider the binary relation for "less than" with the symbol <. This relation does not define an…

A: Reflexive relation: A  relation S on a set A is called reflexive relation if every element of set A…