Define the relation on the set Z of integers by if and only if a ~b if and only if a = b + 5k for some integer k. Show that is an equivalence relation on Z. 3.
Q: , In each item below, verify that - is an equivalence relation on the given set. (b) Let - be a…
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Q: Let B = {a, b, c}and S is the relation on B such that S3 (а, a), (а, b), (b, с), (а, с), (с, с).…
A: Given the set B={ a,b,c } and the relation S on B such that S={(a,a),(a,b),(b,c),(a,c),(c,c)}
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A: 11. Let A be a nonempty set and ~be an equivalence relation on A. (b) We know that for each a, b∈A…
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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 .
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A: It is Irreflexive.
Q: In each item below, verify that ~ is an equivalence relation on the given set. Let S = Z × N. Define…
A: Since you have asked multiple questions, we will solve the Question Q (a) and (b) for you. If…
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: 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: A relation R is defined on Z by xRy if and only if 8 divides 3x + 5y. Prove that R is an equivalence…
A: This is a question from relation.
Q: 3. Let R be the relation defined on Z by aRb if and only if 5 | (4a + b). Prove that R is an…
A: We have given relation on ℤ and defined by aRb if and only if 5|(4a+b). We can see here for any…
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: efine the relation ~ on the set N of positive integers by a~b if and only if a = b(10^k) for some…
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Q: for an equivalence relation R on A, both sets, the class of equivalence classes of R —A/R— is a set.
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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: b
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Q: Prove that "is similar to" is an equivalence relation on Mnxn(F ).
A: We have to prove that “is similar to” is an equivalence relation on Mnxn(F ) i) Every matrix A is…
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: Let R be a reflexive relation on A. Show that R" is reflexive for all positive integers n.
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Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})2: y/x = 2k for some integer k}.…
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Q: (b) Let - be the relation on C defined by z ~ w if and only if z + w e R. Proye that is an…
A: This can be solved as follows:
Q: Q7. Let R be a relation on the set Z defined by the following rule: for all a, b e Z, a Rb if and…
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Q: 3. Define the relation R on Z by mR, if and only if 2|(m – n). Show that R is an equivalence…
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Q: 2. Let R and S be equivalence relations. Prove that RnS is an equivalence relation.
A: Let A be a non-empty set. A relation R on A is said to be equivalence relation if it is reflexive…
Q: 1. Determine whether the relation R on the set of all people is reflexive, symmetric, antisymmetric,…
A: solution
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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: Let A be a nonempty set. Let ~ be a relation on ℘(A)defined by: Letting B and C be arbitrary…
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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: Prove that the relation is an equivalence relation. 1. For x,y E R say x is congruent to y modulo Z…
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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 X be the set of all subsets of ℤ . Define a relation R on X as follows: Given two subsets of…
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Q: Let S be the following relation on R\{0}: S = {(x, y) = (R\{0})² : y/x = 2k for some integer k}.…
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Q: Consider the relation on Z×(Z\ {0}) defined by (m,n)R(m’,n’) provided that mn’ = m’n. Prove that…
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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: Let z be the set of integers and R be the equivalence relation on ZxZ defined by: (a.b)R(c.d) if and…
A: Given that R is a relation on Z×Z defined by (a,b) R(c,d) if and only if a+d=b+c.
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 the set of all integers such that aRb if and only if a + b is even. Is R…
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Q: 6. (a) Let ~ be the relation on Mat2(R) defined by: A ~ B if and only if the (2, 1) entry of A – B"…
A: This problem can be solved as follows:
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: Determine which of the folowing relations R on Z are oquivaknee relatinns. If R is not an equivakner…
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Q: - Let and x be relations on Z defined as follows: For a, be Z, a~b if and only if 2 divides a +b. •…
A: According to the given information, it is required to solve the question 10.
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: Consider the relation R on the set Z defined by a R b if and only if 3|(a+2b) for a, b ∈ Z. Show…
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Q: On the set {(a, b)} of all ordered pairs of positive integers, define (x1, Y1) ~ (x2, Y2) if and…
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Q: Prove that the given relation is an equivalence relation, and describe the distinct equivalence…
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Q: Let R be the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) ER…
A: A ={ (a,b) : where a and b are positive integers} R= {(a,b) and (c,d) : a+d= b+c } We have to…
Q: Let ∼ be the relation on Mat2(R) defined by: A ∼ B if and only if the (2, 1) entry of A − BT equals…
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- 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.True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
- In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each. 10. if and only if .Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.
- In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.In Exercises 610, a relation R is defined on the set Z of all integers, In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and least four members of each. xRy if and only if x2y2 is a multiple of 5.23. 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.