) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b) List of four members of [-2]

Q: c) For a set X = {1,2,3, 4, 5}, let P be the partition {{1,3, 4}, {2}, {5}}. In the equivalence…

A: Q1(c) asked and answered.

Q: 9.32. A relation Ris defined on the set A = {a+bv2: a,b € Q, a + bv/2 + 0} by x R y if æ/y E Q. Show…

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Q: (a) classes of X induced by Prove that - is an equivalence relation on X and describe the set X/ ~…

A: This is a problem of relation.

Q: 1. Let H = {1, 2, 3, 4, 5} and the rlation RC H², with (a, b) ER + a = b( mod 3). • Give the set R.…

A: R ={(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}

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: SULU Let R be a relation over the integers (Z). Prove that R= {(a,b) : a mod 4 = b mod 4} is an…

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Q: 1. Find all equivalence relations on{1,2,3}.

A: Given set is, 1,2,3

Q: Need help trying to solve this

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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: 8) Let A = {2,3,4, 5}, R is defined on A such that (aRB) if and only if (a2 +B+a) is an even number.…

A: Given A={2,3,4,5} We define a relation R on A by (αRβ) if and only if (α2+β+a) is an even number…

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: 3. Prove that if a reflexive R (on some set A) satisfies x Ry A xRz → yRz (1) for all x, y, z, then…

A: the relation is equivalence if it is reflexive, symmetric and transitive.  the relation is…

Q: {0,1,2,3,4,5,6,7,8,9} by x ~y A x2 = is an equivalence relation on X and identify the equivalence…

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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: . 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: 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: Show that the number of equivalence relation in the set {1, 2, 3} containing(1, 2) and (2, 1) is…

A: To show:The number of equivalence relation in the set {1, 2, 3} containing (1, 2) and (2, 1) is two.

Q: 9. Let R be the relation defind on NxN by (m)R(m2) if and only if m1 – n1 = m2 – n2. Describe the…

A: The equivalence class of an element means a set of all those elements that are related to that…

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: (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: From the given information. “S” is an equivalence relation on {0,1,2,3}X{0,1,2}. The equivalence…

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…

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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: Suppose A = {-4,-3,–2,–1,0,1,2,3,4} and R is defined on A by aRb a² - b² is divisible by 4. Prove…

A: A relation R on a set A is said to be an equivalence relation, if it satisfies the following three…

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: 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: 6. Consider the partition P = {{0}, {-1,1}, {-2,2}, {-3,3},{-4,4},...} of Z. Describe the…

A: Given the partition of Z is P=0,-1,1,-2,2,-3,3,-4,4,....

Q: Let A = {2,4,6,8,10}. The distinct equivalence classes resulting from an equivalence relation R on A…

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Q: Let R be the relation of congruence modulo 5. Which of the following equivalence classes are equal?…

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Q: Let A = {1,2,3,4} and let R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,3),(3,3),(4,4)} . Show that R is an…

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Q: Show that the relation R in the set A = {1, 2, 3, 4, 5} given byR = {(a, b) : |a – b| is even}, is…

A: Given that: A = {1, 2, 3, 4, 5} R = { (a,b) ; |a – b| is even}For any element a ∈A, we have |a -a| =…

Q: Give an example of two equivalence relations R and S on the set A = {1, 2, 3} such that RUS is not…

A: 6 Given set is A=1,2,3. Consider the relation R and S on A is given by  R=1,1,2,2,3,31,2,2,1 And…

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: 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: 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: 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: Which of the following are equivalence relations? If they are not, which of the axioms fails? a) For…

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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: 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: 7. Let A = {1,2,3,4}x{1,2,3,4}. Define an equivalence relation ~ by (x1,x2) ~ (x3,xa) iff xxx2 =…

A: Let A=1,2,3,4×1,2,3,4. The equivalence relation ~  is defined by,        "x1,x2~x3,x4 only if…

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: (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: Let A = {1,2,3,4} and let R = equivalence relation. Determine the equivalence classes.…

A: Given A=1,2,3,4 and R be the relation defined by R=1,1,1,2,2,1,2,2,3,4,4,3,3,3,4,4. We have to show…

Q: #2. Let S be the set of all strings of O's and 1's of length 3. Define a relation R on S as follows:…

A: Given is a set S of strings.   Also, given a relation R of set S.   To Prove: R is an equivalence…

Q: (b) Let R and S be two equivalence relations on X. Are RnS, RUS, R \ S, also equivalence relations…

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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: Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are…

A: Given, Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a…