Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is an equivalence relation. (b) Describe the elements in the equivalence class [3]. (a) Show that

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

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Q: Assume that you know that ~ is an equivalence relation on R\ {0} defined as follows: xy > 0. Find…

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A: Given the relation R on ℤ-0 defined by         x,y∈R if and only if xy&gt;0.

Q: Let f: X → Y and define x~y if f (x) = f (y). Show that - is an equivalence relation on X.

A: The objective is to show that ~ is an equivalence relation on X.

Q: Consider the set K rational numbers) defined by x Ry if and only if E K, for x, y E Q+, is an…

A: K=2n:n∈ℤ A relation R is said to be an equivalence relation if it is reflexive, symmetric and…

Q: Let R be a relation on the set of all integers such that aRb if and only if a +b is even. Is R…

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Q: Let R be the relation defined by x R y =x<y° Determine if R is an order relation on N? on Z? Justify…

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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: Let R be the relation on the all integers given by xRy iff x- y 2k for some integer k. (1) Prove…

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Q: A relation R is defined on the set of integers by: aRb = a + b = 2m + 1, where m is an integer. Show…

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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: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

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Q: On sets, the relation x R y given by there exists a bijective function from the set x to the set y'…

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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…

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Q: Let R be a reflexive and transitive relation on A. Show that RN R¯' is an equivalence relation -1 on…

A: First let, A= set of integers. Then, let, R={(a,b):a,b in A} so,

Q: a Define a relation R on N by (a,b) e Rif and only if EN Which of the following properties does R…

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Q: (b) Let - be the relation on C defined by z ~ w if and only if z + w e R. Proye that is an…

A: This can be solved as follows:

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: Consider the binary relation R defined on (N* x N*) by: V (X1. Yı). (x2. Y2) E N* x N*, (x1. Yı) R…

A: Order relation (definition)-: A relation R is called an order relation if it is reflexive,…

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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: 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…

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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: 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: 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: 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: 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: How many distinct equivalence classes exist in the relation R defined as below: x Ry + 3| (2x - )

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Q: . Define a relation on R²\{(0,0)} by letting (x₁, y₁)~(x₂,Y₂) if there exists a nonzero real number…

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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: 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: 4|(x+3y)

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Q: . Let R be the relation on N defined by xR y if x and y share a common factor other than 1.…

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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…

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Q: Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

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Q: Provide a rule to define a relation R on N that is reflexive and transitive but not symmetric. Prove…

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Q: A relation R is defined on the set of all integers as xRy if x - 4y is greater than 5. Then the…

A: We have given that  , R is a relation defined on a set of all integers as x R y  if  x - 4y greater…

Q: Let R be a relation on the set of all integers such that aRb if and only if a + b is even. Is R…

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Q: , 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…

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Q: On the set {(a, b)} of all ordered pairs of positive integers, define (x1, Y1) ~ (x2, Y2) if and…

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Q: On P2, define a relation p(x) ~ q(x) if p(0) = q(0). (a) Show that this is an equivalence relation…

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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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Q: 1. Let A = Z and R, = {(x,y):x² -y is even number} be a relation on A. That is xRy if and only if x²…

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