**Problem Statement: Relationship on Integers** Let \(\sim_R\) be the relation on \(\mathbb{Z}\) given by \(a \sim_R b\) if and only if there exists \(c \in \mathbb{Z}\) with \(\gcd(c, 17) = 1\) such that \(a \equiv c^2b \pmod{17}\). **Tasks:** (a) Show that \(\sim_R\) is an equivalence relation. (b) Identify the equivalence classes of \(\sim_R\).

Q: Explain why each of the following binary relations on S = {1, 2, 3} is not an equivalence relation…

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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: Let ~R be the relation on Z given by a ~R b if and only if there exists ce Z with ged(c, 17) = 1…

A: A relation which is reflexive symmetric and transitive.

Q: Consider a relation R on Z- {0} defined by the rule that (x, y) E R if and only if xy > 0. a) Prove…

A: Given the relation R on ℤ-0 defined by         x,y∈R if and only if xy>0.

Q: Consider the relation being described by the following arrow diagram. Which of the these statements…

A: (i) It is given that ;      S = {6, 7, 8}  and  T = {2, 4, 6} And the relation between S and T is :…

Q: 2 Explain why the relation on N defined as is not an equivalence relation. {(x,y): +y is a multiple…

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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: Let R be an equivalence relation on A = {0, 1, 4, 5, 7, 9} and R = {(0, 9), (9, 0), (0, 0), (9, 9),…

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Q: on P(X), where X = Y, Z E P(X), we say Y - Z if and only if Y has the same number of ele- Define a…

A: Given a set X={1,2,3,4,5,6}. Define a relation ~ on P(X) as : For Y,Z∈ P(X), we say  Y~Z if and only…

Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

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Q: Please help me out

A: Equivalence Classes: The set S is divided into equivalence classes by ~, which is an equivalence…

Q: Let R be the relation on the all integers given by xRy iff x- y 2k for some integer k. (1) Prove…

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Q: Prove relation B is an equivalence relation

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Q: Decide the equivalence classes of the equivalence relation R defined on Z by aRb if 3 | (2a + 7b),…

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Q: Define a relation R on Z by declaring that xRy if and only if x^2 ≡ y^2 (mod 3). (a) Prove that R…

A: (a) for any x in Z,       obviously xRx, as x2 = x2 mod 3       Hence, R is reflexive.        If…

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}?

Q: 5) Let R be a binary relation over the positive integers defined as follows: R = {(a, b) | 2b < a <…

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Q: 1. In the R × R field explain why the definition of (x1, y1)~(x2, y2) if and only if x1 y2 = x2 y1…

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Q: 1. Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of…

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Q: Let the relation R on the set Z be given by the condition: (x, y) E Rif and only if x4 = y4. Answer…

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Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: 4. Let R be the relation on Z defined by xRy if x² = y² (mod 5). Prove that R is an equivalence…

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Q: Lex 172,3,4). Consider the relation R on X defined as follows. R = ((1.1), 13), (2,2), (2.4), (3,1),…

A: I have solved both subpart using basic definition of equivalence relation and equivalence class.

Q: 3) Let S be the equivalence relation on P({0, 1, 2, 3}) defined by XSY if and only if ged(X], 4) =…

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Q: B Let A = {(x,y) |x-yeQ for x, y = R} be a relation on K. Which statement best describes problem?…

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Q: Define a relation R on the set of natural numbers defined by (z, v) E R#z-yis divisible by 3. a)…

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Q: Let R, S, and T be sets. Let f: R -» S, and g: S -» T be maps. Assume we know that qf is 1-1 Must f…

A: Given: