Let i be a positive integer and A; =[1,i+1[. Then U A;= O [-1,1] O R O 11, [ O [1, [ None of these

Q: 10) Let a and b be positive integers. Prove that (a, b) = a if and only if a divides b.

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Q: Let i be a positive integer and A =] - i+1,1]. Then N1 A O 10,1[ O 10,11 O 10,1] O None of these

A: ]0, 1] is correct

Q: Prove that the following statement is true for all positive integers. 1 1 1 + + 4 8 1 2" – 1 + 2n Pn…

A: Given; Pn : 1/2 + 1/4 +1/8 +...+1/2n =(2n -1)/2n

Q: . Let a be a positive number. Prove that for each real number x there is an integer n such that na <…

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Q: Let a be and integer and b be a nonzero integer. Prove that alb if and only if Mb C Ma.

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Q: How to prove that a formula A is valid iff ¬A is unsatisfiable?

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Q: Prove that |ab|= |a | |b| for any numbers a and b.

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Q: 3. Prove that for all x > 0 and all positive integers n, x3 et >1+x + 2! 3! n!

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Q: Given a positive integer n, let a2,n–1 a2,n a3,n-2 a3,n-1 a3,n An = аn-1,2 аn-1,n-2 аn-1,п-1 @n-1,п…

A: Given: Here the given positive integer n, Let An = 00...00a1,n00...0a2,…

Q: Let i be a positive integer and A, = [-i+1,1]. Then U, A O R O 1-0,1] None of these -1,1]

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Q: Let a, b be nonzero integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

A: Given  Let a, b be nonzero integers  To Prove that gcd (a, b) = 1 if and only if gcd(a + b, ab) = 1…

Q: Prove, for all positive integers n, 1 3 2n – 1 1 2 4 2n 3n

A: we can prove given identity by mathematical induction method: Algorithim: Obtained P(n) and…

Q: Find an integer n such that U(n) is isomorphic to Z2 ⊕ Z4 ⊕ Z9

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Q: Let i be a positive integer and A; =] -i+1,1[. Then N,A;= O 10,1] None of these O 10,1[

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Q: Let i be a positive integer and A; =] - i+1,1 [. Then N A= None of these O 10,1] O 10,1[ [0,1[

A: Ai = ] -i+1, 1 [  A1 = ] 0, 1 [ A2 = ] -1, 1 [ Thus ∩Ai = ]0, 1[

Q: Let i be a positive integer and A; =[-i+1,1]. Then U A;= %3D O 1-0,1[ O [-1,1] O R O 1-0,1] O None…

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Q: 14. Let (a, b) = 1 and r = y® for some integers r and y. Then prove that there exists an %3D integer…

A: As per policy, we are solving only the first Question please repost 2nd Question to get its…

Q: Find U, A; and NË, A; if for every positive integer i, a) A; = {-i, -i + 1, ..., -1, 0, 1, ..., i–…

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Q: 4. Let R be the set of all real numbers, xER and n EN such that n 2 2. If (1+x) > 0, prove that Pn:…

A: Let ℝ be the set of real numbers, x∈ℝ and n∈ℕ such that n≥2. If 1+x&gt;0, we want to prove that  Pn…

Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = {n €Z|n…

A: If y is an even integer then it will be of type 2n for some n in integer If y is an odd integer then…

Q: Let i be a positive integer and A; =[1,i+1[. Then U1 A;= O 1, 0 [ O [-1,1] R O None of these O [1,…

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Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: For all positive integers a, b, where a < b, if gcd(a, b) = 1, then a b. O True O False

A: " Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: 1. Prove that for every positive integer n there exist positive integers a11, a21, a22, a31, a32,…

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Q: Proposition 82 (Ex4) Prove: For each integer a, a = 1. if a? = a, then a = 0 or %3D

A: Let a∈ℤ, ℤis the set of all integers. Let a2=a subtract a on both sides ⇒a2-a=a-a ⇒a2-a=0…

Q: Suppose that k and n > k are fixed positive integer. Justify the identity n n + 1 %3D k +1, j=k

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Q: Let k and n be two positive integers, and assume that n is odd. Prove that there exists an integer a…

A: Given: Let k and n be two positive integers in that, we assume if n is odd.

Q: Prove: | ab | = | a | * | b | for all a's and b's in the reals.

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Q: If x and y are two positive real numbers, then there exists a positive integer n such that ny>x

A: Proof: Suppose there does not exist any positive integer n such that ny&gt;x then ny≤x   ∀n∈ℕ

Q: Give an outline for a proof of "for any integer r and any positive integer d, if d|(x– 1), then d…

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Q: 3. Let n e Z be nonzero and even. Prove that there exist unique a € {0, 1}, b e N, and odd c e N…

A: Consider the given condition,  n∈ℤis a non-zero and even. The objective is to prove that there exist…

Q: Let a be any positive number. Prove that |x| &gt;a if and only if x &gt; a or x &lt; -a.

A: Suppose

Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: Label each of the following statements as either true or false.     Let a be an integer. Then (a, 0)…

A: Given :          The statement is ''Let a be an integer. Then (a, 0) = a.''   To determine the…

Q: Let i be a positive integer and A; = {..., i – 6, i – 4, i – 2, i, i + 2, i + 4, i + 6, ... }. Then…

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Q: Let i be a positive integer and A = [-i + 1 ,1]. Then UA,

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Q: Let n, a, and b be positive integers such that ab = n. If (a, b) = 1, prove that there exist…

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Q: Let i be a positive integer and A; =] – i + 1,1 [. Then N, A;= O [0,1[ Ø O 10,1[ O 10,1] None of…

A: Intersection of sets which are forming ascending chain(i.e. A1 contained in A2, A2 contained in A3,…

Q: 2) Prove that o(g

A: To prove that, ogn∩ωgn is empty set.

Q: 1. Let a and b be natural numbers, where a, b > 2, such that gcd(a, b) = 1 and ab = (x + y)" for…

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Q: Let i be a positive integer and A, = [-i+1,1[. Then N14= %3D

A: We have to find the first intersection first.

Q: Let i be a positive integer and A, =1-i+1,1]. Then A O 10.1] O [0.11 O 10.1 None of these

A: Given :- Ai =] -i+1, 1] and  to find:-  ∩i=1∞Ai  We need to find the set of intersection of all Ai's…

Q: {p € Z :x | p}U{p € Z:y|p} C {p € Z :n| p}. : n

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Q: Let i be a positive integer and A; = [1,i+1[. Then U A OR O [-1,1] O [1, [ O 11, [ None of these

A: [1,inf[ is correct answer

Q: (a) Let a,b, and c be integers such that a | b and a | c, and let x and y be arbitrary integers.…

A: According to the guidelines I am answering first question only. a) a, b and c be integers such that…

Q: Prove: Let a,b,x and y be integers. If a|x and b|y then (ab)|(2xy).

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Q: Let i be a positive integer and A; = [1,i+1[. Then U2, A= O [-1,1] O R O 11, [ O [1, [ None of these

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Q: Let n be a positive integer. Prove that for all x € Z, if x" is odd, then x is odd.

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Q: Let i be a positive integer and A, = [-i+1,1[. Then N2, A,- None of these O 10.11 [0,1[ 10.10

A: