Q: Fix a set X, and let S be a collection of equivalence relations on X. Is s U an equivalence relation…
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Q: 8. Consider the binary relation R defined on (N* x N*) by: V (x1. Y1). (X2. Y2) E N* x N*, (x1. Yı)…
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Q: 4. Consider the following binary relation p on the plane R² (x, y)p(u, v) if y? – a² = v² – u².…
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Q: ). Let R be binary relation on N defined by xRy if and only if x- 2s ys x+2. Is R reflexive? Is R…
A: We will usethe definition of reflexive, symmetric, antisymmetric and transitive realtion to answer…
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. if R and T are anti-symmetric relations on a set A then (R U T) is anti-symmetric relation on
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Q: Let R be a relation on the set of all integers such that aRb if and only if 3a – 5b is ev 1) Is R…
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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: Let P be a set on which a binary relation < is defined such that, for all r, y, z € P, (a) * <x is…
A: Consider the provided question, Since, A relation R that is reflexive, anti-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: 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 a relation on the set of all integers such that aRb if and only if a +b is even. Is R…
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Q: R is the congruence modulo 7 relation on Z: For all m, n E Z, mRn iff 7|(m – n). (i) Is the relation…
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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: 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: Let A be the set A-(1, 2, 3} The binary relation R={(1,2), (2,1), (1,3), (3,1), (1,1), (3,3) }…
A: Given: A = 1, 2, 3 and R = (1, 2), (2, 1), (1, 3), (3, 1), (1, 1), (3, 3).
Q: 2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2)…
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Q: 10. (A is the relation defined on Z as follows: for all x, y = Z, x Ay ⇒x=y (mod 3). Describe the…
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Q: Let A = {0,1,2} and r = {(0,0), (1,1), (2,2 y that r is an equivalence relation on A.
A: The relation is reflexive, since a,a∈R, where a∈A Also, since a=b where (a,b)∈R therefore, the…
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: 4. Suppose Rand S are symmetric relations on a set A. Prove that RoS is symmetric iff Ro S = So R.…
A: Suppose R∘S is symmetric Let s, r∈S∘R ⇔r, s∈R∘S⇔s, r∈R∘S ( since R∘S is symmetric) So, R∘S=S∘R
Q: 5. Construct a smallest binary relation S defined on the set {w,r, y, z} such that S satisfies all…
A: Consider the set, A=w,x,y,z, where all the 4 elements are distinct.
Q: (1) Let R be the relation defined on Z by aRb if 2a + b = 0 (mod 3). (a) Prove that R is an…
A: (a) The relation between a and b represents that 2a+b is a multiple of 3.
Q: Let R be the relation on the set N0 of natural numbers given by the following rule: (n,m)∈R if and…
A: Definition of - (1) A binary relation is an Equivalence Relation on a non-empty set S if and only…
Q: 9.45. A relation Ris defined on Z by a Rb if 3a + 56 = 0 (mod 8). Prove that Ris an equivalence…
A: 3a + 5b ≡ 0 (mod 8)
Q: 6. Consider the relation R on Z defined by xRy iff x – y= 4n for some n Ɛ Z. (a) Show that R is an…
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Q: on K³defined by: (a, b, c) {0} such that (a, b, c) = k(c, e, d) is an equivalence relation on K3,…
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Q: Let R be the relation on the set of natural numbers given by the following rule: (n,m) ∈R if and…
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Q: (2) Let R be binary relation on N defined by rRy if and only if r <y< 2r. Is R reflexive? Is R…
A: We have given that R be a binary relation on ℕdefined by xRy if and only if x≤y≤2x. Now we have to…
Q: 9. E is the binary relation defined on Z as follows: For all m, n E Z, m Enm-n is even. Is this an…
A: Equivalence relation
Q: Let R be an equivalence relation on A - {a,b,c,d) such that a Rc and bRd How many distinct…
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Q: on K³defined by: (a, b, c) ~ (d, e, f) if and only if ak e K – {0} such that (a, b, c) = k(c, e, d)…
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Q: . The relation R on Z defined by a Rb if a? = b² (mod 4) is known to be an equivalence relation.…
A: Consider the given information.
Q: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )
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Q: Let R be a relation on Z defined by R = {(p, q) E Z × Z |p – q is a multiple of 3}. (a) Show that R…
A: Since you have asked a question having multiple subparts, we will solve the first three subparts for…
Q: 1 Recall that for x, y ≤ R, |x − y| ≤ Rf is the absolute value of x - y. That is, if x-y≥0 |x - y =…
A: Please see the below picture for detailed solution.
Q: 1. Given the binary relation R on R defined by „R, if and only if x > y – 1, for each of the…
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Q: A relation < is defined on R by x < y if and only if there exists k E N such that x = y + 3k. Prove…
A: Partial order relation
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: Consider the binary relation R = {(x,y),(x,z),(z,x),(z,y)} on the set (x,y,z}, which one of the…
A: We have to find which one of the options is true for relation R.
Q: 1. if R and T are anti-symmetric relations on a set A then (R U T) is anti-symmetric relation on А.
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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: Construct a smallest binary relation S defined on the set {w, x, y, z} such that S satisfies all of…
A: The objective is to determine the smallest binary relation S defined on the set <w,x,y,z> such…
Q: 5) Let R be the relation on the integers where a Rb means a² = b². a) Is this an equivalence…
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Q: (c) Let ~ be a relation defined on Z by a R b if and only if a’ = b³ (mod 4) . Determine the…
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Q: 4. Define a binary relation R on the set of integers Z by (a, b) E R if and only if |a – b| < 1. Is…
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Q: 5. Prove or disprove: for any set A, there exists a relation R on A such that R is both symmetric…
A: Symmetric : If aRb , then bRa , for every a , b∈A Antisymmetric : If aRb and bRa , then a=b
Q: Define a relation T on R as follows: for all x and y in R, x T y if and only if x2=y2. (a) Prove…
A: Given that a relation T on R such that for all x and y in R, xTy if and only if x2=y2 (a) We have to…
Q: 5. Let R be a relation defined on Z by a Rb if and only if 3 | (a + 2b). (a) Prove that R is an…
A: A relation to be an equivalence relation must satisfy the following three properties: 1. Reflexive…
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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- 23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.Prove Theorem 1.40: If is an equivalence relation on the nonempty set , then the distinct equivalence classes of form a partition of .
- Prove that if f is a permutation on A, then (f1)1=f.29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.