The counting measure on (X, A) is defined by μ(E): for all E E A. Set E₁ Then a. Card (E) if E is finite, if E is infinite, = {1,2,3}, E₂ = {4,5,6} and E3 = {3,4,5}. x+ μ(E₁ UE₂) = 6. μ(E₁ UE3) = 6.

Q: Suppose that |A−BI 0.Prove that A=B. HINT: Suppose that A ̸= B and set ε = 1/2 |A − B

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Q: Suppose that E1 and E2 are measurable sets. Prove that m(E1 U E2) + m(E n E2) = m(E1) + m(E2).

A: We have to prove that the given statement.

Q: Let X - U[10, 30] and A = {x < 15}. 1. P(A) = 2. E(X|A) =

A: Given: X~U10, 30 and A=x≤15

Q: (-1)" n+1 nEN. E = %3D n

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Q: 2. Compute e for the following: (a) B= (1 2). (b) B = 3= (1 ²). (c) For B as in (a) (resp. (b)),…

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Q: woite x= psoduct x-6,62E; of dlementasy matsices. as a E2 3

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Q: If u = and , then what is the magnitude of 2 + . Round to the nearest hundredth. Answer:

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Q: FL Let b > 0, b 1. Which of the following must be true if y = Check all that apply. hip O = b* In(b)…

A: no need of explanation as it is itself a formula

Q: Here is some more practice on calculating the number of elements in a set. 10. Suppose A and B are…

A: Given ,           AC U          BC U          nA=73         nB=56        nU=150To find → nA' &amp;…

Q: if L(e3t) = then L(S, e34 du) is. S-3

A: Formula:  If Lft=Fs, then L∫0ufudu=Fss. It is given that Le3t=1s-3. By the formula,…

Q: 3. In each part below, we are given a set Br for every k E N. Write down B1, B2, and B3. Also, find…

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Q: Which sets are equal to the set (a, b, c)? {b, b, b, x, x, x, C, C) {b, b, b, b, b, c, c, c, a, a)…

A: What is Set: Set is the language of mathematics. Through set, we can define mathematical expressions…

Q: Define a function S: Z+Z as follows: For each positive integer n, S(n) = the sum of the positive…

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Q: 6. Let (a, b, c) and let Σ* be the set of all words w using letters from Σ; see Example 2(b). Define…

A: One to one function: A function f is one-to-one if all element of the range of f corresponds to…

Q: 4 1 0 Let X be N , ( μ. Σ ) with 1 30 0 0 2 Examine the independence of the following : (i) X, and…

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Q: Find a natural umber N such that if |a – 5| < e <1 then |a² – 25| < Nɛ.

A: According to our company policy, we are supposed to answer one question at a time. For the rest…

Q: Let a, b eR with a < b. Assume that a + <b - for all n E N. Show that b- = (a, b). %3D n n=1

A: We will show  two sets are equal.

Q: Determine the intervals of continuity for the following function. AY 5- 4- 3- 2- 14 0+ 1. 2 3 4 6. 5

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Q: If x = (1.5,2.5), what is the smallest interval I = (4-ɛ,4 + ε) for which we can guarantee that x²…

A: Given that x∈(1.5,2.5) and I=4−ε,4+ε

Q: (d) For which e E {0, 1, ..., 9), is 12e128 divisible by 8? Find all such e. 1

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Q: Let G=(V, E) a graph, then show |{ € V\dg(x) = 1 mod 2}|= 0 mod 2.

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Q: 1 2. Let e =. Determine the smallest choice of N > 0 such that N. 6 ' n(n + 1)

A: To determine the smallest choice.

Q: Given the graph of the polynomial function, determine its equation: 6- 4 2- 0 -2- -4- H 2 3 4 X

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Q: Let a, b eR with a < b. Assume that a + <b - for all n e N. Show that n n=1

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Q: A container contains 35 green tokens, 5 blue tokens, and 3 red tokens. Two tokens are randomly…

A: There are total (35+5)=40 non-red tokens and 3 red tokens in the container.  And total (35+5+3)=43…

Q: et H={Φ∈Sn | Φ(n)=n}. Find index of H in Sn.

A: H=ϕ|ϕ(n)=n=I (group of identity permutation)o(H)=1o(Sn)=n!index of H in Sn=n!1=n!

Q: Page 164, 3.1.15* Let X have the pmf p(x) = (1/5)( 4/5)*, x = 0, 1, 2, 3, . . ., zero elsewhere.…

A: Conditional probability : P(A|B) = P(A ∩ B) / P(B)

Q: Find UBE3 B and NBE3 B for each collection 3. a. S = {[1,1+]:nEN} b. 3= {(1,1+¹):n € N} c. 3=…

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Q: Let (2, F,µ) be a measure space. (a). Let A, B E F such that A C B. Prove that u(A) sH(B). (b) Let…

A: Given that Ω,F,μ be a measure space. We know that Ω,F,μ is a measure space if: μϕ=0…

Q: Let G = (V, E) a graph, then show |{ V|dc(x) = 1 mod 2}|= 0 mod 2.

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Q: (1.2.7)List all elements in the following sets. (a) T = {m € Z | m = 1 + (−1)¹ for some integer i}…

A: The sets are: (a) T=m∈Z | m=1+-1i for some integer i (b) U=r∈Z | 2≤r≤-2 (c) U=r∈Z | -2≤r≤2 To Do:…

Q: Let s = {x €N : 2 < x < 9} and T = {x eN : 5 s x < 14}. Then snT = {x €N : 5s x < 9}. Select one: O…

A: We have to choose the correct option.

Q: 2. Let A = {0,1, 2, 4, 5, 7} and B = {2,3, 4, 5, 8} be subsets of the set %3D %3D U = {n € Z | 0 < n…

A: Given: A=0,1,2,4,5,7  &amp; B=2,3,4,5,8 is subset of U=n∈ℤ  0≤n≤9 i.e.U=0,1,2,3,4,5,6,7,8,9 To Find…

Q: Suppose that µ is the measure defined in item 3. What is µ(O) where O is the set of natural odd…

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Q: Let E = . Determine the smallest choice of N > 0 such that N. |n(n+1)|

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Q: 5. Show that the set {e" : n e Z} is countable. (Hint: Use the function f(n) = en and show it is 1-1…

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Q: A restaurant that is famous for its desserts offers 100 different dessertst Of these desserts, 45…

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Q: Find parametric equations for the line through Po = (10,-1, 1) perpendicular to the plane 2x + 3y -…

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Q: (9) Show that the given collection F is an open cover for S such that it does not contain a finite…

A: (9) (a) Given that S = 0, 2 and F = Un | n∈ℕ where Un = 1n, 2-1n. The objective is to show that the…

Q: Let E-fr.8.2): os es , 05 25 1,7√25 rs 7). Compute 6.8365-1051090 mit Answer x III. r

A: ∬∫Ey2 e(x2+y2)3= ∫0π∫01∫7z7(r sin θ)2. e(r2)3. r dr dz dθ= ∫0π∫01∫7z7(r3 sin2 θ). e(r6) dr dz dθNow,…

Q: (b) Let k E Z. Show that {x E Z | x divides k}C {y €Z ] y divides k2}. (c) Show that the set {x E N…

A: The answers are in the next steps.

Q: Consider a set S={1,2,3,4,5} and P(x)=x−110, where x is a member of S. Calculate E(S).

A: It is given that P(x)=(x-1)/10.