2. Define the relation on S = Z by a~b + a = b mod 5. (1) Prove that is an equivalence relation. (2) Find all equivalence classes of ~.

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: 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: (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: 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: ). 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: 2) Let R be binary relation on N defined by r Ry if and only if r <y< 2r. Is R reflexive? 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: 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: For the function S : Z4s → Za defined as f(z) = 4r + 1 mod 6. Co defined as zRy when f(x) = f(w).…

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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: Define the relation ~ on the set Z of integers by m~n if and only if m^2 = n. Show that ~ is not…

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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: Consider the equivalence relation {(r, y) | a = y (mod 6)}| on the set {n e Z| – 9 <n< 15}| Use the…

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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: How many (distinct) equivalence classes does the relation R-(1.1), (2.2) (3,3),(4,4), (1,2), (2.1).…

A: The relation is R=1,1,2,2,3,3,4,4,1,22,1,3,4,4,3 The set is X=1,2,3,4

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: 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: 5. Let A=Z and R, ={(x,y):x-y is divisible by 5} be a relation on A . That is x R,y if and only if…

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Q: Let D be the relation on the natural numbers defined by the rule m divides n if and only if m does…

A: Let D be the relation on the natural number is defined by the rule m divides n if an only if m does…

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: 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: 7. Recall that, for n e Z, n > 0, there is a relation on Z called congruence mod n. The b (i.e.…

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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: Let R be a relation on Z. Then Ris an equivalence relation if it is defined by xRy if and only if…

A: R is an equivalent relation iff R is reflexive. R is symmetric. R is transitive.

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: 2. Suppose I wish to define a function g : {0, 1, 2, 3, 4} → Z that takes an input of an equivalence…

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Q: Let S be a set of distinct symbols. Show that the relation defined onW(S) in this chapter is an…

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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 ~ 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: a. The relation x = y if and only if x mod 4 == y mod 4 is an equivalence relation. Use this…

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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: Let R be the relation "congruence modulo 5" defined on Z as follows: x is congruent to y modulo 5 if…

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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: The relation defined as x ~ yx ≡ y (mod7) for x, y ∈ Z in Z Write the equivalence classes by writing…

A: Relation R is defined in Z as x≡y(mod 7)  (difference of x and y is multiple of 7) Reflexivity : For…

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

A: A relation R on a set A is said to be an equivalence relation if it is reflexive symmetric and…

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 "congruence modulo 7" defined on Z as follows: x is congruent to y modulo 7 if…

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Q: 3. Prove that a = b(mod7)is an equivalence relation by taking suitable examples.

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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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