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 only if 5 divides a – b. Prove that R is an equivalence relation on Z.

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: 1. Given a relation R on a set A, prove that if R is transitive, then so is R-

A: Transitive means aRb and bRc then aRc

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: Suppose that R is a reflexive and transitive relation on a set A. Define a new relation E on A by…

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

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Q: 4. Let R be a relation defined on the set of integers by a R b if a +b = 0 or a – b = 0. Determine…

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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: 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: 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: Show that the relation R in the set A of all the books in a library of a college,given by R = {(x,…

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

A: Concept:

Q: Let the relation non z defined by x ny if and only if 3/(x+2y) Show that n is an equivalence…

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Q: Given the relation R= {(1,1), (2.2), (3.3)(4.4), (1.2). (2.1) (3.4). (4.3)) on the set X- (1,2.3.4).…

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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: Subject: Set Theory

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Q: Let R be the relation on the set of people with doctorates such that (a, b) ER if and only if a was…

A: Let A be the set of all people with doctorates and R be a relations on A defined by a,b∈R if and…

Q: Let H be the set of all human beings. Consider the relation R defined on H by xRy if and only if x…

A: R defined on H by xRy if and only if x and y have the same biological mother. reflexivity : x &…

Q: 1. For (a, b) and (c, d) in R², define (a, b)R(c, d) by a² – c² = b2 - d². Determine whether the…

A: The given problem is to prove that the relation is equivalence relation, to be equivalence it must…

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: Show that the relation R defined on the set of integers ℤ by (a,b) Î R if a – b = 6k for some k Î ℤ…

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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: A relation R is called atransitive if aRb and bRc implies cRa. Show that R is reflexive and…

A: Given a relation R is called atransitive if aRb and bRc⇒cRa Now shows in the following that R is…

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: et S ⊆ N, and for any a, b ∈ N, consider a relation R such that aRb if and only if there exists c ∈…

A: This is the problem of Relation.

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: 2. Let R be a relation defined on the set of integers Z where a R b if a +b is even. Determine…

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Q: Define a relation Q on the set R × R as follows. For all ordered pairs (w, x) and (y, z) in R x R,…

A: The relation Q on the set R×R is defined as follows. For all ordered pairs w, x and y, z in R×R, w,…

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: ) 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: 9.15. A relation R is defined on Z by a Rbif |a – b| < 2. Which of the properties reflexive,…

A: Given : A relation is R is defined on Z is given by  a R b if a-b≤2  Which of the properties…

Q: Show that the set T := {(x, y) E Z × Z | x+y is even number } is an equivalence relation on the set…

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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 R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: Consider the set Q = Z ×(Z 10), and the relation tupon QDefined as: (a, b) ↑ (c, d) e a·d=b·c a - d…

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Q: A relation on a set A is circular iff , y, z x~y and y~z implies z~x. Prove that a relation is an…

A: The relation on a set is a subset of the Cartesian product of the set. Therefore, each element is in…

Q: A relation * is defined on the set R-{0} by x * y if and only if xy > 0. For each of the following…

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Q: Let R be a relation from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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Q: Define a relation R on the set J ={ 0, 1, 3, 4, ,5, 6 } as follows for all x, y E J x R y if and…

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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: For (a, b),(c,d) EN×N define the relation (a, b) ~ (b,c) if and only if ab = cd. Is this relation…

A: Explanation of the answer is as follows

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 from A to B and S a relation from B to A. Prove or disprove that if S of R =…

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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: 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: 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: 1. Consider the following relations defined on the set {a, b, c}. For each relation, determine…

A: (a) Reflexive functions are the ones whose domain and range are the same  for instance a,a∈R1 So…

Q: Show that the relation R defined on the set of integers ℤ by (a,b) ∈ R if a – b = 6k for some k ∈ ℤ…

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