Use a particular counterexample to explain why R fails to be an equivalence relation is the definition of the subsets is adjusted as follows: A1 = {x|x € Z* and 0 < x < 10}, A2 = {x|x € Z* and 10 < x < 20}, A3 = {x|x € Z* and 20 < x < 30}, ...

Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

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Q: 3. Let the relation R on X = {1,3,5, 6, 8, 9, 11} be such that xRy means "x – y is a multiple of 3".…

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Q: In Q1-Q4 prove why each relation has or does not have the properties: reflexive, symmetric,…

A: Since you have asked  multiple questions so as per guidelines we will solve  the first question for…

Q: Every equivalence relation is reflexive. True False

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

Q: Given a non empty set X, consider P(X) which is the set of all subsets of X.      Define the…

A: The relation is defined as follows.

Q: R —A/R— is a set. You will need, of course, to find a superset of A/R, that is, a class X

A: To prove the given relation, show that as, reflexivity, symmetry, transitivity.

Q: Give an example of a relation on the set {a, b, c, d} which is symmetric and transitive, but not…

A: Any relation R from a set Ato A is reflexive if and only if for every x∈A , the pair (x,x)∈R Any…

Q: 1. Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible…

A: According to question given that Set Z in which the relation R is defined by aRbIf and only if a+3b…

Q: A relationR on a set A is called equivalence if R is:

A: Equivalence Relation: A relation R on a set A is said to be an equivalence relation if and only if…

Q: B if there exists o E Sn so that oao- = B. Prove that For a, B E Sn, let a ~ equivalence relation…

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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: b) Let X = (x, y, z). Then show that A = {(x,x),(y,y),(z,z)} is an equivalence relation. Determine…

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Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

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Q: 4. a) Define the following terms: i) POSETS ii) Equivalence relation

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: {(a, b) acb, a and b are finite set of integers}

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Q: Define the relation ~ on the set Z of integers by m~n if and only if m^2 = n. Show that ~ is not…

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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: Show that the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b} is an…

A: Given, the relation R in the set Z of integers given by R = {(a, b) : 2 divides a– b}.

Q: Consider the equivalence relation {(r, y) | a = y (mod 6)}| on the set {n e Z| – 9 <n< 15}| Use the…

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Q: 1. Let F be a nonempty family of transitive relations. Prove that NF is a transitive relation. Hint.…

A: Given that the F be a nonempty family of transitive relations.

Q: 1. Prove that “divides" is a transitive relation. That is, prove that, for all integers a, b, c, if…

A: I the given question we have to prove that , for all integers a,b,c, if a|b and b|c, then a|c.

Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

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Q: 1. Let & be the relation of (a,b) la #b] on the set of integers. What is the reflexive closure of R?

A: As per our company guideline we are supposed to answer only one question with its 3 subparts. Here…

Q: Without using the ?∞i=1 Ri representation of R+, Prove that for any set relation R (we did not pick…

A: Given: Let us consider the given, R+=∩Q:R⊆Q∧Q is transitive and prove that for any set relation R.

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: Let S' = {x € R² : ||x|| = 1} C X. and endow S' with the same equivalence (b) relation - as above.…

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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: If R1 and R2 are equivalence relations in a set A, show that R1 ∩ R2 is also an equivalence…

A: Given, R1 and R2 are equivalence relations in a set A. So, both R1 and R2 are Reflexive, Symmetric…

Q: Suppose A is a set and R is a reflexive and transitive relation on A. Show that the

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Q: Let A be a set and let ∼ be an equivalence relation on A. Let x, y ∈ A. Prove that [x] = [y] if and…

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Q: Consider the set Q = Z × (Z 10), and the relation fupon Q Defined as: (a, b) ↑ (c, d) + a•d = b•c a)…

A: we shall solve the part (a) only , if you want us to do other parts please resubmit the question…

Q: Let N be the set of all natural numbers and let R be a relation in N, defined by R = {(a, b)} : a is…

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Q: ) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: For nonempty sets A, B, and C, |A| <= |B| and |B| <= |C|, then |A| <= |C|.

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Q: Decide whether the following relations is a partial order or not. (a, b) | a Í b, a and b are…

A: We have to decide whether given relation is partial order or not?

Q: Let us consider the following relation R on the set of all pairs of integers: (a,b)R(c,d) if a = c…

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Q: 4. Prove that the next relation is equivalence relation on the set of all people: {(x, y)|x and y…

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Q: Is the relation aRb → ab < 0 an equivalence relation on the set Z? Justify your answer.

A: Given a relation, aRb⇔ab≤0. To check whether R is an equivalence relation on set ℤ. A relation is an…

Q: Let A be a set and R be a reflexive relation on A. Prove that R is an equivalence relation if and…

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Q: (b) By the Equivalence Relation Theorem, there is an equivalence relation R on Z50 such that P is…

A: A relation from X to X is called a relation in X.  If R is in X and: xRx ∀x ∈ X ⇒ R is refelexive…

Q: A relationR is called a partially ordered set if R is: .a reflexive .b antisymmetric .C transitive…

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Q: Let R be the following relation on the set of all people. R= {(a, b) : a and b share a common…

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Q: Suppose an equivalence relation R has the following equivalence classes that partition the set X.…

A: The ordered pairs in the equivalence relations produced by the given partitions {0, 2}, {1}, {3, 4}.…

Q: Let X = {1, 2, 3, 4, 5} and Y = {3, 4}. We define the relation R on the set P (X) by the formula (A…

A: Here, I have shown that the given relation is an equivalence relation and found out the equivalence…

Q: We defined the relation between sets by A - B means that there there is a 1-1 correspondence f : A →…

A: The complete solutions is in given below

Q: Prove that the given relation is an equivalence relation, and describe the distinct equivalence…

A: Let A is the set of all students at your college. For every x,y∈A,  xSy⇔x is the same age as y. That…

Q: Let 2 be a non-empty set. (a) Show that the inclusion relation in P (2) is an crder relation…

A: A partial order relation leads to poset which is very interesting.