Define a relation R on the set of all integers Z as follows: for all integers m and n, m R n → m = n (mod 3) Prove that R is an equivalence relation.

Q: 1. Given a relation R on a set A, prove that if R is transitive, then so is R-

A: Transitive means aRb and bRc then aRc

Q: , In each item below, verify that - is an equivalence relation on the given set. (b) Let - be a…

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Q: Consider the set A = {2, 3, 5, 11, 12, 13} and the relation Va, b E A, a R b → a = b mod 3. List all…

A: Given the set, A=2, 3, 5, 11, 12, 13 and the relation ∀a,b∈A, a R b↔a≡b mod 3

Q: Given a relation R on a set A, prove that if R is transitive, then so is R-1.

A: Suppose A be any set and R is relation defined on A. Then R is transitive if whenever (a,b)∈R and…

Q: Let R be a relation on the set of all integers such that aRb if and only if 3a – 5b is ever 1) Is R…

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Q: Let R be a relation on the set N of positive integers defined by aRb if and only if the product axb…

A: This question is about equivalence  class .

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: Consider the relation K on N defined by Knm iff n <m.Which statement below is true? K is an…

A: It is Irreflexive.

Q: 1. (p.209, #16) Define a relation R on Z by declaring that xRy if and only if x? = y? (mod 4). Prove…

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Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers . If    x…

A: x = y  and     O &lt; z

Q: Let A = {82,95, 146, 153, 158, 166, 180, 187, 207} and R be an equivalence relation defined on A…

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Q: Let S be the set of integers. If a, b E S, define aRb if a + b is even.Prove that R is an…

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Q: SULU Let R be a relation over the integers (Z). Prove that R= {(a,b) : a mod 4 = b mod 4} is an…

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Q: consider the relation ∼R on P(N), the power set of N, with A  ∼R B iff A∩B ≠ ∅ determine whether…

A: consider the relation ∼R on P(N), the power set of N, with A  ∼R B iff A∩B ≠ ∅

Q: Define R on Z+ by aRb if = 5 for some integer k. (a) Prove R is an equivalence relation. (b) What is…

A: The solution is as follows:

Q: Let R be the relation on the set of ordered pairs of positive integers such 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: . Define r on the power set of {1,2, 3} by ArB → |A| = |B|. Prove that r is an quivalence relation.…

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Q: Define a relation R on Z by declaring that xRy if and only if x^2 =y^2 (mod4). Prove that R is…

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Q: Let R be the relation on A × A , where A= {1,2,3,4} that is elements of this relation are pairs of…

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Q: Define the relation on the set Z of integers by if and only if a ~b if and only if a = b + 5k for…

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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: Consider the equivalence relation ~ on R given by a ~ b if and only if [a] = [b]. You do not need to…

A: Given that the equivalence relation ~ on ℝ defined as a~b if and only if a=b. Define a function…

Q: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

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Q: b

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Q: F is the relation defined on Z as follows: Vr, n E Z, mFn =7|(m² – n²). (a). Prove that F is an…

A: Equivalence relation and classes

Q: Subject: Set Theory

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Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers . If x…

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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: Show that each of the following relations equivalence classes. on Z is an equivalence relation, and…

A: In the given question we have to find the equivalence classes.

Q: Let N be a nonempty set and o a permutation of N. Define a relation a~b if and only if b = o"(a) for…

A: Given, Ω is a nonempty set and σ a permutation of Ω. A relation on the set is defined as a~b if and…

Q: (3) In each item below, verify that - is an equivalence relation on the given set. (c) Let - be a…

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Q: 9.51. Let R be the relation defined on Z by a R b if 2a + 3b = 0 (mod 5). Prove that R is an…

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Q: Let ~ be defined on the set X = {0,1,2,3,4,5,6,7,8,9} by x ~ y + x2 = y? mod 5. Prove that is an…

A: Equivalence relation are those which are reflexive symmetric and transitive

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: 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 R be the relation on the set of integers such that a R b if and only if a = b or a = - b.…

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Q: 1. Determine whether the relation R on set Z (set of integer number) is reflexive, irreflexive,…

A: We determine whether the relation R on the set Z is reflexive, irreflexive, symmetric, antisymmetric…

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: Let S be the congruent modulo 7 relation on Z, that is m Sn 7|(m-n). Show the relation S is: (a)…

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Q: Let A = {-5, -4, −2, 0, 3, 6, 8), and define an equivalence relation R on A as follows: (x, y) E R…

A: Let A = {-5, -4, -2, 0, 3, 6, 8}, and define an equivalence relation R on A as follows: (x, y) in  R…

Q: Define - on Z as follows. Suppose that a ~ b if a² = b² (mod 6). Prove that - is an equivalence…

A: We need to prove 1) Reflexive 2) symmetric 3) transitive.

Q: #1. Let R be the relation defined on the set of all integers Z as follows: for all integers m and n,…

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Q: Let R be the relation on the set N0 of natural numbers given by the following rule: (n,m)∈R(n,m)∈R…

A: Let R be the relation on the set N0 of natural numbers given by n,m∈R if and only if n-m is a…

Q: #2. Let S be the set of all strings of O's and 1's of length 3. Define a relation R on S as follows:…

A: Given is a set S of strings.   Also, given a relation R of set S.   To Prove: R is an equivalence…

Q: Define a relation R on the integers Z saying that (m, n) is in R if m2 is equivalent to n2 (mod 7).…

A: First to prove the relation, R, on set of integers Z is equivalence relation. R is defined as:

Q: 3. Prove that a = b(mod7)is an equivalence relation by taking suitable examples.

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Q: Let R be the relation on the set of integers defined as aRb + 5a + 8b = 0 (mod 13). (a) Show that R…

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Q: 8. Let A be a set of nonzero integers and let R be a relation on A × A defined by (a, b)R(c, d)…

A: Equivalence relation

Q: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…

A: Prove the understated statements concerning the relation &lt; on the set Z of all integers .   If O…