Show that both ℝ and ℝ2 may be covered by countably many open balls

Q: Let ő1, 82 E R such that 0 < 61 < 82. Show that B(p, 81) C B(p, 82). Suppose S and T are balls with…

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Q: 3.2 Prove that in any metric space (S, d) every closed ball S,xo is a closed set.

A: Let (S, d) be a metric space. Consider a closed ball Srx0 with both center x0∈S and radius r&gt;0…

Q: 3.3 Let r1 and r2 be distinct points in the metric space (S, d). Verify that there are open balls S,…

A: 3.3 is also known as Hausdorff Property. d(x1, x2) &gt; 0 for every x1 ≠ x2   (metric space…

Q: Calculate | [(ydx – xdy)] wherein y(t) (2cos(t), 2sin(t)),0 < t < 27, is an circum ference that…

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Q: Let X N (u,1) of size n. Obtain UMVUE for M.

A: We will use Lehmann-Scheffe theorem to find the uniformly minimum variance unbiased estimator for μ.

Q: Why is F= {An = (1/n, 1), nE N;n>1}not an open cover of [0, 1]? Which points in [0, 1] does UAn not…

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Q: Prove that the open ball B(a,r) is a subset of the interior of the closed ball C(a,r)

A: Given that Ba,r is an open ball of radius r and centered at a. Then a∈Ba,r                ...... (1)…

Q: 2 Let A | and U = H. Then A2017U is equal to = ... A) B

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Q: Consider the Venn diagram below. Illustrate the other De Morgan's law by selecting the region(s)…

A: GivenAc ∪ Bc

Q: Prove that the set S = {n² : n e N} is denumerable. %3D

A: We have to show that the given set S is a denumerable set.

Q: Show that S {x€R* |2,• 24. • O} is/isn't %3D convex

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Q: 4. Assuming that the c possible subset of R (c) f(x) = V5 –

A: Use the definition of domain and range to solve this problem

Q: Consider the Venn Diagram below. A В 2 7 6 9 %3D п(В)- n(AN B)= n(A U B)= | n(A)= %3D n(B)=

A: Solving the venn diagram problem by solving the values.

Q: Given a 2 0 and b > 0, let u = a and v = Vab Use the Cauchy-Schwarz inequality to compare the…

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Q: Prove by mathematical induction: X n (U-, Y) = U=1(X nY), for V n E Z* – {1} and sets X,Y,,Y2,. Yp-…

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Q: 4).ェ Pare subset, -f y such that CeD than fic) is not eD, then the Convers %3D そイue.

A: Solution As it's mention C=D and f^-1 same , so each and every element from  f inverse we get equal.…

Q: 2. Prove that there are no rectangles in S2. More precisely, prove that if A, B, C, D E S² form a…

A: Given: ∠A=∠B=∠C=∠D=π2

Q: A, is skew-symmetric, then prove that (B B + B, B') An = . Ajk = 0. %3D

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Q: Let X ∼ Bin(n, p), and let Y = n − X. Show that Y ∼ Bin(n, 1 − p).

A: Here X follows Binomial with parameters n and p. Here, Y=n-X. Then Y takes the values n-0, n-1,…

Q: Show that if A and B are bounded subsets of IR, then inf(A U B) = min{inf A.inf B}. %3D

A: To show  If A and B are bounded subsets of R then inf(AUB)=min{infA .infB}

Q: How many points need to be in a set in R° to guarantee that the set is affinely dependent? А. 11 В.…

A: According to the given information, it is required to tell the number of points in a set R9 to…

Q: 4. Let b > a > 0. Show that (a,b] is neither closed nor open.

A: Explanation of the solution is given below....

Q: Which of the following sets are open, and which are closed? Explain your answers by sketching the…

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Q: A designer is creating various open-top boxes by cutting four equally-sized squares from the corners…

A: Given that, a square of sides x length is cut from each corner of the sheet. Then the length and…

Q: The number of non-empty subsets of B is |{S:SC A, S = 3 and Sn B = 0}| = max {|S|: SC AUB, SNA| is…

A: Given that A and B are finite sets satisfying A=6, B=7, A∩B=2. Since, B=7. So, total number of…

Q: - Show that the area of the triangle whose vertices are the [x1 yi det X2 Y2 A - points (x₁, y₁),…

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Q: Consider the collection of sets {Bn : n E N}, such that Vn EN, B, = {x € R : -n <a <n}. Find U B,…

A: We are given, So, B1 = { x ∈ R | -1 ≤ x ≤ 1},       B2 = { x ∈ R | -2 ≤ x ≤ 2},       B3 = { x ∈ R…

Q: Show the following se ts Convex ? are for ast sh} k k-1 PCt)=a,+aztr...a t

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Q: Let d1, 82 E R such that 0 < d2 < 82. Show that the d1-ball B(p, 81) is a subset of 52-ball B(p,…

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Q: (b)* Show that Y' is closed.

A: Actually we shall show, closure of the set = the set. Then this set will be closed.

Q: 4. Let A %3D {a, b, c}, В %3D {x, у}, and C %3D {0, 1}. Find a) АхВх С b) СхВхА с) СхАх В d) ВxВxВ.

A: Given that: 4. A={a, b, c}, B={x, y}, C={0, 1}   NOTE: As per our answering guidelines, we can only…

Q: Show that the sete S = {x E R]x = m/3, where m E N with m > 4} is countable

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Q: Let V = {cities of Metro Manila} and E = {(x, y) |x and y are adjacent cities in Metro Manila}. %3!

A: Given: Let v= cities of Metro Manila  and E=x,y/ x and y adjacent cities in Metro Manila The…

Q: Consider the word TORTOISE. Count the number of arrangements of this word (a) such that no two…

A: Consider the given word TORTOISE. The objective is to count the number of arrangements of this word:…

Q: 5-The Center of aringR dented by cent CR)is the set Cent CR) - [CERIC.X = X-C,V XER? %3D Prove that…

A: solution You just need to prove that the centre is a ring within itself. Associativity of both…

Q: 3. Show that X = (x|Ax=b, x ≥ 0} is convex. following ure which no

A: As per our guidelines we are supposed to answer only one question.

Q: Given two red rooks and t ue rooks, find the number of distintt woys in which the nooks Can be…

A: it is given that there are two red and two blue books. And there is a board that has, n×n cells.…

Q: In R4, let S = span{(1, 1, 1, 1)T ,(1, 2, 1, 3)T }. Find S⊥.

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Q: Find the intersection of the sets. {w, y, z}∩∅

A: Given, w,y,z∩∅we have to find intersection of the sets.

Q: Prove that the set S = {n² : n E N} is denumerable.

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Q: Let d, a, b E N. Prove that

A: Given,  d, a, b∈ℕ To prove that if d|a+b2  and   d|a-b2, then d|b.

Q: (b) Using Venn Diagram to prove that (A U B)e = A° N B° ?

A: To prove the following using Venn diagram A∪Bc=Ac∩Bc First consider the left hand side  Set A

Q: c) Compute µr, Hy, Ož, o, Cov(X, Y), and p

A: Reviewing information, Given pdf is, f(x,y)=x+y                 ; 0≤x≤1 ,  0≤y≤1

Q: (2) Let S := {1,..., n}. Compute the number of towers of length of the form A1 c A2 c S, where A1 +…

A: A tower of length  set is a basically chain of subsets of different size.

Q: Let D be the closed region in

A: Given,                    x2-2xy+y2+x+y=0   and      x+y+2=0

Q: Let n=23 and consider the following information, Model One: Y = Bo + B1X1 + B2X2 + B3X3 + B,X4 +e…

A: Solution: Given information: n= 23 observation: k= 3 independent variable  R2=0.8685

Q: -. Show that both R and R2 may be covered by countably many open balls.

A: Soution.

Q: 3.3 Let 11 and ra be distinct points in the metric space (S, d). Verify that there are open balls…

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