Question 2: Show that the set S = {rVT + V2:rE Q) is dense in IR. Explain and show your work.

Q: Let G = Z31, S = {1,5, 6, 25, 26, 30}. %3D %3D Prove that Cay(G, S') is normal.

A: It is given that

Q: Question 24 Prove that if SRS, = {X1, X2, X3,...,X} € N(u, 0), then X = N (, ) Edit View Insert…

A: Given that; Xi  follow the normal distribution with mean μ and variance ?2 . By using sampling to…

Q: Chapter 15, Section 15.1, Question 018 Find div F and curl F. F (x, y, z) = xz" i + 6y"x² j+ 3z²yk…

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Q: Q.1 a- Verify Euler's theorem U= x"In(?)

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Q: Prove that if h E L'(R) and [ h= 0 for all s E R, then h = 0.

A: Use the Leibniz integral rule: For ∫axbx fx, tdt Where -∞<ax, bx<∞ We have, ddx∫axbx fx,…

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

A: GivenAc ∪ Bc

Q: If sup(S) exists, then it is unique ylgn ihi

A: The solution is given as

Q: Given the bases B={u , u2} and B'={u'1, u'z} for R2, where , U2 and u'2 = The transition mat rix…

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Q: Let TE L(V, W), where V and W are finite-dimensional. Prove that rk(7) rk(7*).

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Q: Prove that on average, permutations of length n have H, cycles, where Hn i=1

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Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

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Q: Let yk = k² and Zk = 2k|k|. Are the signals {yk } and {Zk } inearly independent? Evaluate the…

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Q: Chapter 15, Section 15.1, Question 017 Find div F and curl F. F (x, y. 2) = x° i- 7j + yz k Enter…

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Q: Given the bases B = {u1, u2} and B' = {u'1, u'2} for R?, where and u'1= The transition mat rix Pg-B'…

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Q: Theorem 4.1. if r, s ER and r <s. Then there is q € Q with r <q <s (i.c. Q is dense in R)

A: n,s∈R and n&lt;s Now, s-n&gt;0. By Archimedean property of R, there exists a natural number n such…

Q: Is S = {a + b√(2) : a, b ∈ ℤ} dense in ℝ?

A:

Q: Question 1: Prove the following identifies: a) A-V AV.

A:

Q: Given: ZB ZD Prove:A ABC -A EDC в Ma com A D

A: Explained below...

Q: Prove that on average, permutations of length n have H, cycles, where Σ Hn i=1

A: We will take an arbitrary set containing n elements and observe the required permutations.

Q: Show that every positive sol fron of Ay + B, k=osl,2- & bosunded, %3D JK is i-e, ogy sB-if A<1. 1-A

A: Given yk+1=Ayk+B, k=0,1,2,... Then the objective is to find the solution at which the sequence is…

Q: QUESTION 1 Let f(x) = In x (1,e4) on Find both Upf) and Lpf) for the partition P= {1,e',e?,e³,e*}

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Q: A natural cubic spline S on [0,2] is defined by |So(x) = 1 + 2x – x³, S(x) = if 0<x< 1, |Si(x) = 2 +…

A: Given Sx=S0x=1+2x-x3if 0≤x&lt;1S1x=2+bx-1+cx-12+dx-13if 1≤x≤2

Q: Given the bases B= {u,,u2} and B'= {u'1, u'2} for R, where U2 and The transition mat rix Pg-g from B…

A: We first find the coordinate vectors of u_1 and u_2 relative to basis B' ,then we will form the…

Q: A natural cubic spline S on [0,2] is defined by (So(x) = 1+ 2x – x³ S(x) = Is, (x) = a + b(x – 1) +…

A: Given: Natural cubic spline S on [0,2] is defined as: Sx=S0x=1+2x−x3 if 0≤x≤1S1x=a+bx−1+cx−12+dx−13…

Q: Let a,b E R with a # 0, b # 0. Prove that (ab)¯1 = a°1b´1. %3D

A:

Q: If z, w E C, prove that z+w=z+w.

A: This follows from simple substitution. The detailed solution is presented in step 2. As requested i…

Q: Given the bases B = {u,u2} and B'= {u', u'2} for R?, where and u'= The transition mat rix Pg-g' from…

A: Linear transformation problem.....

Q: Let P = {(z, y, 2) E R° | az + by + cz = 0} where (a, b, c) # (0,0,0). (a) Prove that (a, b, c) ¢ P.…

A: Given that P=x,y,z∈ℝ3 : ax+by+cz=0 where a,b,c≠0,0,0 (a) The objective is to prove that a,b,c∉P Let,…

Q: Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x,g (y) = 3y + 4 and h (z) =…

A: Given, Consider f : N → N, g : N → N and          h : N → R defined as f(x) = 2x, g (y) = 3y + 4 and…

Q: Problem 9. Prove that if f : [a, b] → R is integrable on [a, b] then so is |f| and

A:

Q: 8. (a) Suppose f : [a, b] → R is integrable and L(ƒ, P) = U(ƒ, P) for some partition P of [a, b].…

A: please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Let z, w E C. Show that zw = 0 implies z = 0 or w = 0. %3D

A:

Q: a. Show that T: 1² → 1² defined by Tx = y = (n), n, = r, is compact where x = (5,) and where0 <r< 1.

A: Show that T: l2→l2 defined by Tx=y=ηj, ηj=rjξj is compact where x=ξj and 0&lt;r&lt;1. Definition:…

Q: Define ƒ : [0, 1] → R by f(x) = = x. Prove that L(P, f)<SU(P, f) for every partition P of [0, 1].…

A: The given function is f:0,1→ℝ defined by fx=x

Q: Question 9. Let A E Mn,n. Prove that A is symmetric if and only if Ax · y = A"x · y for all x, y E…

A: See the solution it is done step by step.

Q: 4. A region D in RN is said to be convex if and only if for every u, v E D and t e (0, 1), then tu +…

A: Given: A region D in ℝN is said to be convex if and only if for every u,v∈D and t∈0,1, tu+1-tv∈D.…

Q: Given: AB = AC, RB = RC %3D Prove: SB = SC %3D B A- R

A: Given÷ AB=AC,RB=RC PROVE÷ SB=SC

Q: Let A:e? e be gioen ACX,の2-10 2=10, prove thet A Find the spectrom of A. is compact Jun

A: Given linear operator A:l2→l2 is defined by: Ax1,x2,…,xn,…=0,x12,x222,…,xn2n,…. (a). We Know that…

Q: Find the of the GS that satisfies the given Cond hb Sum a n=4 b. h=6

A: Sum of Geometric series.

Q: QUESTION 3 Prove: ( N A = U A°. a EA Attach File Browse Local Files

A:

Q: Question 2: ( {(x,2x,3x):x E R}. Let E %3D (a) Show that E is not bounded. (b) Is E is compact? Why?

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Q: QUESTION 1 Compute [[ ev** where Q is bounded by y = V100-x-z AP 2++ and y = 0.

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Q: Let SCR and let c e R be a cluster point of S. Prove that for each e >0 the set W := Sn(c-e,c+e) has…

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Q: Exercise 4.11 Suppose m is Lebesgue measure and A is a Borel measurable subset of R with m(A) > 0.…

A: Given, m is Lebesgue measure and A is a Boral measurable subset ofℝ with m(A)&gt;0.  To prove that…

Q: A subset W of R° is given by = {(x,y, z) | 2x = y = 2z}. (i) Write down the distinct elements of W.

A:

Q: Evaluate the following: 4 1. ;dA where T is the region enclosed by the triangle (x2 – 4x + 7)2 with…

A: As per the company guidelines we can solve first question. I hope that the given solution will help…

Q: Let X1, X2, - ,X, be iid N(0, o²). Consider the hypothesis test : Ho : a? = 1 H : o² = 4 Let C1 =…

A: Let X1,X2,...,Xn be iid N0,σ2. Consider the hypothesis test: H0:σ2=1H1:σ2=4 Let C1=s'2≥K1, C2=s2≥K2…

Q: QR I RS and UV 1 TÚ. Construct a proof that ZQRS = ZTUV.

A: First, we sketch the situation roughly  We have to prove:

Q: Consider a S = {x|x"=1} is S, equals to S?

A: