Let a₁ ≥ a2 >. ≥ an ≥ 0 and b₁,b2, ... ers such that k k [a; ≤[b;, j=1 j=1 ≤k ≤n. Prove that

Q: Define the function fn : [0, 1] –→R for n > 1, defined as if æ E (,1] nQ if x E (, 1]\Q if x E [0,…

A:

Q: 3. Let f : [2,5] → R such that f(x) et Is f uniformly continuous? Justify your answer. Answer:

A:

Q: J Assume E₂ and E₂ one measurable. Without using the fort that the Union of measurable sets in…

A: Given that  and  are measurable.We need to prove that  is measurable.We know that .We know that, if…

Q: Let S=X belongs to R: x> =1. a) Show that the set S has lower bound, but no upper bounds. b) Show…

A: The answer is given as follows:

Q: Give the condition which ensure that Je-| < 1 where z E C.

A:

Q: Let f(r) = 4e and g(x) = In(5z): find (f o g)(2). %3D a) O 60 b) O 100 c) O 16 d) O 40 e) O 400 f)O…

A:

Q: 2. Let f: ZxZ-ZxZ be defined by f(m, n) = (3m+n,n²) a. Is fone-to-one? b. Is fonto?

A:

Q: Let X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

A: Solution

Q: Let x, y ∈ R such that x 0.001. Prove that |x · y| = |x| · |y|

A: Given: x, y∈ℝ, x&lt;-0.001 and y&gt;0.001   To prove: x·y=x·y

Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

A:

Q: Show that S= {xER*| x < 2x.} IS convex

A:

Q: Question If f: (R, mL) → R and g: (R, mL) → R are measurable then:

A: Given statement:-  If  f  :  ℝ, mL  →ℝ  and   g  :  ℝ, mL  →ℝ  are measurable then:

Q: 23. Let A be a non-empty set of positive numbers, which is bounded above and let B = {xe R: 1/xe A}.…

A:

Q: Let f(x, y) = x(y - 3) and F = Vf. If r = with 0 ≤ t ≤ 2, then [F. dr is equal to: a. -7 b. -11 c.…

A: We find the line integral in the following way

Q: Let A,BE Mmna Shaw that vTAw= V' Bw for all VER™ end WEIR" if and only if LA=B. %3D

A:

Q: Prop (1.23 ≤ x ≤ 1.72) = Prop (-2.50 ≤ x ≤ 0.5) = Prop (x ≥ 3.00)

A: here use normal distribution table

Q: If 4 p and q P крса are conjugale exponent a measure X a measure space, p р. f₁g:x [₁₂+] measurable…

A: Given P and q are conjugate exponent and f,g are measurable function and let p,q∈(1,∞)

Q: y = C¡e e-2= + Cze³* + C2e%

A:

Q: 1. Let A=(a)xn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. |a|< (aa)², i,…

A:

Q: 2. Let a € R be given. Define f: R" → R by f(x) = a¹x, x = R¹. Using the € - & definition, show that…

A:

Q: Queston #o 4 Show that X, Ixe, where x, and Xe E IR X= (1,1,1, -1) Xz = (4,2,2, -1) %3D

A:

Q: Let f(x) ={ 2+ Vex-x , for os x 2 } 2 4 evaluate fCx) dx and I f6) dx

A: Given piece wise function as, f(x) = 2+2x-x2    for  0≤x≤24-x                  for  x &gt;2we need…

Q: Find Pr (|w| < 0.2)  where W ~ N (-1,4)

A: From the given information, W follows normal distribution with mean µ=-1 and variance σ2=4.

Q: Let A = {x ER : 2 < x < 3} and B = {r €R: -1 <r< 2}. Prove that |A| = |B|.

A:

Q: Exercise 5. Let (E,T) be a measurable space, let (m,)ısisn be n measures on (E,T), and le (a;)ısisn…

A: Let E,τ be a measurable space and mi1≤i≤n be a n measure on E,τ.  Suppose, ai1≤i≤nbe real positive…

Q: Let a₁,..., an EC with |aj| < 1 for all j = 1,...,n and A₁,..., An ≥ 0 real numbers satisfying 1 j =…

A:

Q: 2b. In R show that : || x - y || || x + y || ≤ 11x11 ² + 11y11²2² -(1

A:

Q: On the measurable space (R, B (R)), let u: B (R)→ R be defined as: μ(A) = The number of all rational…

A: Given: We are given a measurable space , where represents the collection of Borel sets on the real…

Q: 2) Let a,bE Q such that a<6. Prove that ax <bx.

A: The statements can be proved by analyzing the results from the given statements. The other methods…

Q: Let f(x)= x+1, find all xeR such that f[f(x)]=2 the solution set is: а. у%3D0. b. y=x. с. у#х. d.…

A:

Q: If X is uniform on [2, 6]. Find a. P(X > 3), b. P(4 ≤ x ≤ 5)

A: X~U(a,b) probability density function pdf of uniform distribution is.  f(x)=1/(b-a)  , a&lt;x&lt;b

Q: For each q € Z+, let B, denote the set of points in [0, 1] whose image under f is at least 1/q: B₁ =…

A: Let's denote the set Bq as defined in the question: \[ B_q = \{x \in [0,1] : f(x) \geq…

Q: Let S be the set of integer solutions to the equation ri + r2 +13+x4 = 30, where ri >0 for i = 1,2,…

A:

Q: Let A = {y ∈ Z : 2<y ≤ 14 and y is even}. What is |A|?

A:

Q: 1 ifrE (, 1]nQ 2 if re (1\Q 5 if re (0,)nQ 6 if re(0,1)\Q 0o if r= Ja(z) = a) Prove that f, is a…

A: Since you have put a multipart question, as per company policy, we shall answer only the first part.…