, Let Z be the set of integers. Consider R = {(x, y) e Z x Z: 7 divides x-y} Z x Z. Show that R is an equivalence relation on Z.

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: Let A C B and let S be an equivalence relation on B. Prove that R = {(x,y) E A × A : (x, y) E S} %3D…

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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 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)}

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

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Q: Let m e Z+. Show that =m is an equivalence relation on Z. Prove that for any integer a,

A: Equivalence relation

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

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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: 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: b. Define a relation SER XR by S ={ (x,y) E R × R |x – y € Z }. Prove that S is an equivalence…

A: By equivalence relation, we mean that the relation is reflexive, symmetric, and transitive.Here we…

Q: that P ◦ Q is an equivalence relation on X iff P ◦ Q = Q ◦ P

A: Concept:

Q: Show that if R and R2 are equivalence relations on A, then R1 n R2 is an equivalence relation on A.

A: Equivalence relation on a set A

Q: Define the relation R on Z as follows: for x, y ∈ Z, xRy if and only y−x ≥ 0. Determine whether R is…

A: partial order means reflexive and antisymmetric.

Q: Consider the relation R on R2 given by for any (a,b),(c,d)∈R 2, (a,b)R(c,d) if and only if a 2 +b…

A: According to the given information, Consider a relation R on R2 as:

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: 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: Let T be the relation defined by, For all a, b e Z, aTb iff a – b is odd. Is T an equivalence…

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Q: Let X = R? – (0,0) and - be defined over X by setting X~y + y = tx for some non-zero t e R. (b)…

A: Sol

Q: Consider the set Q = Z × (Z 10), and the relation fupon Q Defined as: (a, b) ↑ (c, d) + a•d = b•c a)…

A: we shall solve the part (a) only , if you want us to do other parts please resubmit the question…

Q: 4. For r, y E R, let z y if and only if (x- y) E Q. Show that defined as such is an equivale…

A: ~ is defined as : For all, x, y in R, x~y if and only if (x-y) belongs to Q Reflexivity : x~x for…

Q: ) Let S = {(x, y) E R × R: y – x is an integer} a) Prove the S is an equivalence relation on R b)…

A: Given, S=x,y∈R×R: y-x is an integer

Q: Let R be a relation defined on Z as follows: (a,b) is an element of R if and only if the square of a…

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Q: Define a relation R on Z by xRy → x = y + 5k for k E Z. Prove that R is an equivalence relation by…

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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 whether the following statements are true or false: b) The relation R defined by xRy7[(x-y)…

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Q: Let S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an…

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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: 1. A relation R is defined on Z* × Z* by (m,n)R(p, q) → m+q = n + p . (a). Prove that R is an…

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Q: Let C = {a, b, c, d}. Find a relation R on C that has exactly 3 ordered pair members and is both…

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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: 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: Let S be the following relation on ℂ: S = {(x, y) ∈ ℂ2 : y − x is real}. Prove that S is an…

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Q: A relation R is defined on R by a R b if a – b e Z. Prove that R is an equivalence relation and…

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Q: 8. Let A = Z* U {0}, define relation R on A × Z* by (k,l)R(m,n) → kn = lm. (a). Prove that R is an…

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Q: Let S be the following relation on C: S = {(x, y) = C²:y-x=bi for some real number b}. Prove that S…

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Q: RUR and RNR symmetric relations on X. are Show that R=Z xZ is an equivalence relation.

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Q: 14. For a,b E Z, define a ~ b if and only if 4a + b is a multiple of 5. Show that defines an…

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Q: Let G={(m,n)∣m,n∈N x N and gcd(m,n)=1}, and define the relation q on G according to (a,b) q…

A: A totally ordered set is a set G together with a relation ≤ that satisfies the following conditions.…

Q: Let R be a relation defined on Z by xRy if and only if x-y=7k for kez. The equivalence class of [5]…

A: Given relation is xRy iff x-y=7k So, equivalence class of 5 will contain ....-9, -2, 5, 12, 19,....…

Q: In this exercise, you want to show that a relation RC A? is transitive + R" C R for all n EN=…

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Q: Define a relation R on Z by declaring that xR y if and only if x^2 ≡ y^2 (mod 4). Prove that R is…

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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: Let a relation R on R2 be given by (x,y)R(x’,y’) provided that x-x’ and y-y’ are integers. Prove…

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Q: Let S be the following relation on C10). S=[(x,y) E (C\[0]): y/x is real). Prove that S is an…

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Q: Let S be the following relation on C: S = {(x, y) = C²:y-x is real}. Prove that S is an equivalence…

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Q: Let R and S be reflexive relations over set A. Prove or disprove that RnS is reflexive

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Q: For a,y E R, let z~y if and only if (x - y) € Q. Show that ~ defined as such is an equivalence…

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