True or False 1. If {An} is a monotone non increasing sequence of sets then U₁A₁ = A₁. 2. zand z partition the set of all integers. 3. Let A = {0, 1, 2,..., k} where kEZ*. Then lim A, as k→∞ does not exist

Q: True or False? 11. If Γ = {[0, 1/n) | n ∈ N} then ∩Γ = [0, 1). 12. Let R be a relation on a set A.…

A: 11) if Γ ={[0,1n) | n∈N}  then ∩Γ=[0,1)         Γ= {[0,1), [0,1/2), [0,1/3), [0,1/4), [0,1/5),…

Q: Determine the set ⋂ n=1 to ∞​ (−1−1/n,1+1/n) (intersection of open intervals)

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Q: for i = 1,2, 3. Suppose also that P(X = i) = for i = 4, x. 1,2, 3. 8. Suppose P(X,n = i) = 3n+6…

A: A sequence X1,X2,X3,...Xn,... be a sequence of real valued random variables is said to be…

Q: 27. Let A be a set and let f: A A be a function. For x, y E A, define x ~ y if f(x) = f(y). (a)…

A: This is a problem of set and function.

Q: lim (1,9) >(2,4) || O 12 O 15 09 O does not exist O 6

A: Please refer the attached image for complete solution.

Q: 8. Let a₁ ≥ a2 ≥ ... ≥ an and b₁,b2,..., bn be two sequences of non-negative real numbers such that…

A: The given problem is to prove the given inequality for the given two sequences an and bn and…

Q: 10. Prove that (n) ≤ n/2 for n ≥ 8.

A: To prove that π(n) ≤ n/2 for n ≥ 8, where π(n) represents the number of prime numbers less than or…

Q: Which 4-tuples a 0<a<b<c<d<6}. are int the relation {(a, b, c, d)|a, b, c, d are integers with

A: We need to identify the tuples of the integer relation .Let .

Q: 9. Let a, b eR such that a a} I. (а, 5) A. I and II only В. П only C. I and III only III. (-00, b)…

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Q: 10. Show that the interval (0, 1) is covered by the open intervals In (1/n, 1- 1/n), n 2 3, but (0,…

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Q: Let A = {2" : m e Z}. A relation - is defined on the set Q+ of positive rational nu a a ~ b E A is…

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Q: The relation by a = b (modn) is Partial Order Relation Ture O False O

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Q: [3 marks] Let A = (-∞, 5], B = [-1,4), and C = (3, 8]. Then, BnC is O(3, 4). Op. O[-1,8]. O[3, 4].…

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Q: Let < be a partial order on A. Assume that a,b EA are elements with the property that: for any X EA,…

A: (First part): Reflexive property : Since ≤ is an reflexive relation then for every element x∈A we…

Q: is lim x = inf (xn) as x, y. What Express answer and = 3n lim sup (xn) where Xn = [(-1)³ ]+2?

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Q: Q. 3. A transformation T:R² →R² first reflects vectors through the y axis, and then rotates vectors…

A: Answer 3: We are given a transformation T: R2→R2 that first reflects vectors through y-axis and then…

Q: Let D be the set of all finite subsets of positive integers. Define a function T:Z* → D as follows:…

A: Since T(n)=The set of positive divisors of n. Since 17 ,5 are prime numbers.

Q: Q1. Let A be a set contained in a given Hilbert space H. State the conditions for A to be…

A: According to the policy, we are solving only first question only. Please resubmit it and specify…

Q: 4. For each positive integer i, let A, = {x € R| -<x < } = (-;;4) a. Find U-, A;. i=1 b. Find N 4:-…

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Q: QUESTION 1 Is relation R on set of all integers reflexive where (x,y) ER if x is a multiple of y?…

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Q: 6. A relation R on a set is called transitive if whenever ( a,b) is an element of real numbers and…

A: Q6: True Q7: option (c) is correct.

Q: 5. Using the limit method, show that every polynomial p(n) = a¿n² + ak- belongs to O(n*). ... + ao,…

A: Big theta notation is an order notation. It determines the growth rate of two growth functions. It…

Q: {(-)} 3. k 1

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Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: 11. A one-to-one correspondence is a function which gives an exact pairing between the elements of…

A: In mathematics, a bijection, bijective function, one-to-one correspondence, or invertible function,…

Q: Question 12: If sin 0 + cos 0 = /3, then %3D prove that tan 0 + cot 0 =1.

A: Introduction :

Q: (2.1) Let b> 0 and X [-b, b). Here we admit %3D B"open" in r if and only if (0} B or (-b, b) C B.…

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Q: 1. てー d Sinlt).losl). 1-(-1)2 ー1 1. Suppore an an dao 十} an and a, =2." Show that 9nt! the Sequence…

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Q: 3. Suppose (a,n) is Cauchy and that for all m, k 2 100, Jam - as 0. Does it follow that the…

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Q: . Is the power set P(H),

A: Given,   And the powerset of H: . * and *' on are defined by . We have to justify that the…

Q: k* for some constant k". Lp.) Show that L(po)

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Q: Suppose poset (A, R), where A = {1, 2, 3, 4, 6, 8. 9. 12, 16, 18, 32, 36} and the relation R runs…

A: With the help of definition we solve the given problem as follows:

Q: - Give an example of a sequence of positive rationals {a„}, such that the following statement holds:…

A: Let us consider a sequence of positive rationals {an}∞ n=1 defined as a1=1  and  for n≥1  a2n=an+1…

Q: 21. Prove that if n and k are natural numbers such that k < n, then (*:) - (-) - () n k k

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Q: 4   1. Define the relation | on N\{0, 1} by a | b precisely when a divides evenly into b.   (b)…

A: Given relation I on N-{0,1} fora,b∈N-{0,1} a∣b⇒b=La for l∈ℕ ∵a∣a  ∀a∈N-{0,1} ∴ I is Reflexive.