State whether each of the following statements is true or false. Then give an explanation, proof or counter example to justify your answer, The distinct equivalence classes of S form a partition of B. S is a partial order relation. If r and y are elements of B such that r y and rSy, then (9.2) S.

Q: (3) Let S be the equivalence relation on (0, 1, 2, 3} x {0, 1, 2} defined by (a, b)S(c, d) if and…

A: S=0,1,2,3×0,1,2=0,0;0,1;0,21.0;1,1;1,22.0;2,1;2,23.0;3,1;3,2

Q: ) Let R = {(a,b) : 3keza = k * b} be a relation over the positive integers. Prove or disprove that R…

A: Given that R=(a,b):a=k*b To show reflexive it is enough to show that a~a. It is true because, a=1*a.…

Q: Let A = follows: For V(a,b), (c,d) = A x A, (a,b)R(c,d) ⇒ a ≤ c and b ≤ d {0,1,2} and the partial…

A: a. The Hasse diagram is shown below.

Q: Let R be a binary relation R on Z, defined by: Vx, y € N*, x R y =3neN* such that y = x" (a) Show…

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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: Find, for each part of the question, an example of a set S and two relations R1 C S x S and R2 C S x…

A: Since the questions contains many subparts. As per the guidelines I am answering the first three…

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: 9.24. Let R be an equivalence relation on A = {a, b, c, d, e, f,g} such that a Rc, c Rd, d Rg and b…

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Q: Let R= {(a, b): 3keza k* b} be a relation over the positive integers. Prove or disprove thatR is a…

A: A relation R is partial order relation if R is reflexive, antisymmetric and transitive.

Q: Below is the Hasse diagram for a partial order relation R on the set {a, b, c, d, e}. e d be C Are…

A: Introduction: Hasse diagram is very significant in graphing a relation. In this diagram, the…

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

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Q: Let Z be the set of integers. Consider R = {(x, y) e Z x Z: 7 divides x-y} S Z x Z. Show that R is…

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Q: Let S ⊆ N, and for any c, d ∈ N, consider a relation R such that cRd if and only if there exists a ∈…

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Q: - b if and only if f (a) = f(b). Show that ~ is an Let f: A → B be a surjection. Define a ~…

A: Prove ~ is an equivalence relation. 1) Reflexive Since f is a function, f(a) = f(a) and so a ~ a for…

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: Let R be a relation on A = {1, 2, 3, 6} and R = {(1,6), (1, 1), (3, 3), (6, 1), (1,2), (2, 1), (2,…

A:

Q: State whether each of the following statements is true or false. counter example to justify your…

A: Note: Since you have posted a question with multiple sub-parts, we will solve first three sub-parts…

Q: Let S ⊆ N, and for any x, y ∈ N, consider a relation R such that xRy if and only if there exists z ∈…

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Q: Consider a relation ~ on the set Z such that x~y if and only if x + y is even. The relation ~ is…

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Q: Let D be the binary relation defined below. For all u, w e 2, u D w → w=u +7 or w= u - 7. Show…

A: Solution of given problem is as below: A binary relation is equivalance relation if it satisfies the…

Q: a) Prove that R is an equivalence relation. b) Find [C] for C = {1, 3}. ) How many different…

A: Follow the answer below.

Q: (b) Observe that all the relations in (a) are equivalence relations. Prove that this is true in…

A: Given: A is a nonempty set, and P is a partition of A. To show: The relation R is equivalence…

Q: (i) The equivalence class [b] determined by b E B is a non-empty set. 1 (ii) If r and y are elements…

A: Given that S and T are equivalence relation on a set B. i ) the equivalence class [b] determined by…

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: 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: Show that the relation R defined on the set of integers ℤ by (a,b) Î R if a – b = 6k for some k Î ℤ…

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Q: Let R be a relation on A = {1,2, 5, 8} and R = {(8,5), (2, 1), (5, 5), (5, 8), (2, 2), (8, 1)}. Are…

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Q: Prove or disprove: If R and S are partial orderings on A, then R ∩ S is also a partial ordering on…

A: Partial order relation of intersection of two partial order set

Q: Let R={(a,b)∈R×R: there is an integer k s.t. a−b= 2kπ}. (a) Prove that R is an equivalence relation…

A: (a) Prove that R is an equivalence relation on R. So, we need to show that R is reflexive, symmetric…

Q: et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈…

A: This is the problem of Relation.

Q: 2. Let R = {(1, b) ERX R: there is an integer k such that a – b = 2kn}. Prove that R is an…

A: us the definition of reflexive , symmetric and transitive property of relation to prove it is…

Q: Let R be a partial order relation on Z*. For any positive integers a and b, (a, b) E R if a|b. Are…

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Q: Let A Z and let the relation R be defined on A by zRy if and only if 7 divides r-y. a) Show that R…

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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: 1. A relation R is defined on Z* × Z* by (m,n)R(p, q) → m+q = n + p . (a). Prove that R is an…

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Q: 4) Suppose R ={(a,b) €Z×Z:b-a is divisible by 3}, show that R is an equivalence relation on Z.…

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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: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

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Q: 7.3.4 (a) Let - be the relation defined on Z by a~ b + a +b is even. Show that ~ is an equivalence…

A: Since the second question is independent of the first question I am answering the first question…

Q: 2. Let H ≤ G and define = on G by a = b iff a¯¹b € H. Show that =µ is an equivalence relation.

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Q: Let R be a relation on A = {1, 2, 6, 7} and R = {(6,6), (2,2), (7, 1), (1, 1), (7,7), (1,7)}. Are…

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Q: Which one of the following is not a partial order relation? (a) The relation defined by r | y e y =…

A: A relation R on a set S is said to be a partial order relation if it satisfies the following 3…

Q: (5) Let A={-2,-1,0,1,2}. R is an equivalence relation defined as: for all x,y E A, xRy 2| (x-y?).…

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Q: Which of the following statements is TRUE? sgn( (5 8) )=1 Let R be a relation on Z defined by xRy if…

A: ?Relation R is not Reflexive, because 2+2=4 does not divisible by 5. So , (2,2) not in R. ?…

Q: Show that the relation R defined on the set of integers ℤ by (a,b) ∈ R if a – b = 6k for some k ∈ ℤ…

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Q: Let R be a relation on A = {0, 3, 5, 8} and R = {(0,0), (5, 5), (3, 3), (5, 0), (8, 8), (8, 3), (0,…

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Q: 1. Let A = Z and R, = {(x,y):x² -y is even number} be a relation on A. That is xRy if and only if x²…

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Q: 2.4 (34). Let R be a partial order relation on a finite set S. Show that S has a maximal element…

A: Given: Let R be a partial order relation on the finite set S

Q: Give examples of relations R and S on Z such that (a) Both R and S are not equivalence relations but…

A: (a) Let R and S be two relations on ℤ such that both R and S are not equivalence relations. Let…

Q: 2. Let R be the relation on Z→Z such that ((a,b),(c,d)) = R a+d =b+c. Show that R is an equivalence…

A: