Let S be the set of integers. If a, b E S, define aRb if a + b is even.Prove that R is an equivalence relation and determine the equivalenceclasses of S.

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

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Q: Let R be the relation on the set of integers, s.t., aRb if and only if a=b or a= -b. Is R an…

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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 = be an equivalence relation on a set A. For every element a E A, let [a] denote the equivalence…

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Q: Let R be the equivalence relation on the set A ={ a, b, c, d } induced by the partition ( {a,b,c};…

A: Given A=a, b, c ,dpartitiona,b,c; d

Q: Given a non empty set X, consider P(X) which is the set of all subsets of X.      Define the…

A: The relation is defined as follows.

Q: Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following: (a)…

A: 11. Let A be a nonempty set and ~be an equivalence relation on A. (b)  We know that for each a, b∈A…

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: 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: Show that “is unitarily equivalent to” is an equivalence relation on Mn×n(C).

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

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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: Let A be the set of all statement forms in the three variables p, q, and r, and let R be the…

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Q: Let A be a set and R be a symmetric and transitive relation on A. Prove the following statement: If…

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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 S be the set of real numbers. If a, b e S, define a - b if a – b is an integer. Show that ~ is…

A: Definition of the equivalence relation helps to prove the required. A relation is said to be an…

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: Suppose R and S are symmetric relations on a set A. Prove that RoS is symmetric iff RoS = So R.

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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 ~ 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: b

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Q: Prove that the following relations are equivalence relation: On Z× (Z−{0}), with (a,b)∼(c,d) if…

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Q: Let R = { ( a,b ) | a,b ∈ Q and a − b ∈ Z } . Prove that R is an equivalence relation on Q

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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: In each item below, verify that ~ is an equivalence relation on the given set. (a) Let S = Z x N.…

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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: 9.29. Let R be an equivalence relation defined on a set A containing the elements a, b, c and d.…

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Q: 11. Let A be a nonempty set and let ~ be an equivalence relation on A. Prove each of the following:…

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Q: Show that the relation R on Z defined by aRb if and only if 5a−3b is even, for a, b ∈ Z, is an…

A: Let R be a relation from a set A to itself. R is said to be an equivalence relation if 1) R is…

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: For nonempty sets A, B, and C, |A| <= |B| and |B| <= |C|, then |A| <= |C|.

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Q: Let A={(m,n)∣m,n∈N and gcd(m,n)=1} and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc…

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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: Let L be a language. Let us define the relation R such that x Ry if for every string z we have that…

A: First to prove R is reflexive, For let consider x , then for every z  we have xz∈L↔xz∈L. Hence R is…

Q: Determine, with justification, whether the following relations are equivalence relations. If so,…

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Q: let S be the set of real numbers. if a,b is an element of S define a~b if a-b is an integer,show…

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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: Consider the set Z in which the relation R is defined by aRb if and only if a + 3b is divisible by…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: (b) By the Equivalence Relation Theorem, there is an equivalence relation R on Z50 such that P is…

A: A relation from X to X is called a relation in X.  If R is in X and: xRx ∀x ∈ X ⇒ R is refelexive…

Q: 10. Let be the relation defind on Z by „R, if and only if a|b. Orve an explicit description of the…

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Q: Let A={(m,n)∣m,n∈N2 and gcd(m,n)=1}, and define the relation ⪯ on A according to (a,b)⪯(c,d)⇔ad≤bc.…

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Q: Let R be the following relation on the set of all people. R= {(a, b) : a and b share a common…

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Q: 1. Let X be a set, R, S C X × X binary relations on X. Prove or disprove the following. (a) If R and…

A: We have to solve given problem:

Q: Let R be an equivalence relation on A. Find the maximum number of classes R can make if JA| = 80 and…

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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: 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