4 points). For each of the following relations ~ on the integers Z, say whether ~ is an equivalence relation; explain your answer. If is an equivalence relation, give an explicit description of the set of equivalence classes Z/ ~. a.~y if and only if x ≤ y. b. xy if and only if x - y is even.

Q: Let R be the relation on the set of integers, s.t., aRb if and only if a=b or a= -b. Is R an…

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Q: Let A = {2,3,4,...,17,18). Let R be the equivalence relation on AxA, defined by (a,b)R(c,d) if and…

A: Let A = {2,3,4,.......... 17,18].Let R be the equivalence relation on AXA, defined by (a,b)R(c,d) if…

Q: (24 pts) Either show the following relations are equivalence relations, or show which qualities of…

A: a. S = {0, 1, 2, 3}, R = {(0,0), (1, 1), (1, 2), (2, 1), (2,2), (2, 3), (3, 3)} Reflexive: Yes. We…

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Q: prove why each relation has or does not have the properties: reflexive, symmetric, anti-symmetric,…

A: The given relation R is as follows. R=x,y∈Z×Z:x is a multiple of y If x,y∈R, we denote x related to…

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: • Relation "z + y is divisible by 5" on integers z, y is an equivalence relation. Circle the answer:…

A: A relation R is called an equivalence relation if : R is Reflexive, i.e. aRa  R is Symmetric, i.e.,…

Q: 4. Let R be a relation on the set of all integers. xRy iff x - y = 3m for some integer m. Show that…

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Q: Let R be the relation on the set of ordered pairs of positive intergers such that (refer to image).

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Q: Let A = {a, b, c, d} and R = {(a, a), (b, a), (b, b), (c, c), (d, c), (d, d)}. Determine whether the…

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Q: If a relation is an equivalence relation, state the set of equivalence classes.

A: For the given relations To check the properties (i) Reflexive (ii) Symmetric (iii) Transitive.

Q: Let R be a relation defined on the set of integers as follows: a R b iff 6 divides  (a - b). R is…

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

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Q: 1. For (a, b) and (c, d) in R², define (a, b)R(c, d) by a² – c² = b2 - d². Determine whether the…

A: The given problem is to prove that the relation is equivalence relation, to be equivalence it must…

Q: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.

A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…

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: 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…

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Q: What is the significance of the line above the 'a' variable in this context?

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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: The relation R defined on Z by aRb if 3 | (2a + 7b) is an equivalence relation. the equivalence…

A: The objective of the question is to understand how the equivalence classes for the given equivalence…

Q: Define the relation R on 72 = Z x Z by (a, b) R (c, d) if a + d = b+ c. Prove that R is ar…

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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: 3. Let X be a set and S and R equivalence relations on X. Prove or give a concrete counter example…

A: Since you have asked multiple questions, we will solve the first and second question for you. If you…

Q: b. Let us consider the sets A = {2,3,4,5} and B = { 0,1,2,3}. Define the relation R: A → B such that…

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Q: For the set A = {are, bot, cab, era)}, define the relation R on A to be: (x,y) eR if and only if…

A: As per the condition, in every order pair there should be only 1 letter in common.

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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Q: 6. (a, b), C

A: Let R be the relation on the set of ordered pairs of positive integers such that a,b,c,d∈R if and…

Q: Let the relation R on the set of ordered pairs of positive integers be defined as (a, b) R (c, d) a,…

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