Prove Boole's inequality: 00 00 PJa)

Q: a – b -| < 1, if|a| < 1, and |b| < 1. 1- āb - Show that|-

A: From the given information. For , then show that .

Q: Let A = {x Rx ≥ 5 and x 8 and x < 12} (a) Find (AUB) nC. (b) Find An BNC. (c) Find Ce

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Q: Prove that ||A||1 = maxi<j<n{E#1 laij|}. i=1

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Q: Prove that for any vector x, we have (a) x₂ ≤x≤√₂x₂. 1 (b) x ≤x≤x n

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Q: (b) Let x € R. Prove that if x² – 2x − 3 ≥ 0, then x² + 1 > 0. -

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Q: 1. Let u, ER" prove that lũ ơi <ll||||||

A: we want to shoe that  u→·v→≤u→v→

Q: |u| = |v| show that (u + v) • (v - u) = 0 is logically equivalent to condition u⚫u=v•v

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Q: If X is uniform in the interval [-3,3] compute X(i.e. E{X}) and o² (i.e.E{(X-X)²}).

A: Given that X ~ U[-3,3] to find mean and variance

Q: (а b)с + (а - с)b + (с b)a 3 ve that v belongs to B.

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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…

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Q: Prove that for any x, y ≥ 0, we have √x - √y ≤ √√x - y). (Hint. Consider the cases x≤y and y ≤ x…

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Q: Let u, ER" prove that ||ū + ï|| ≤ ||ū|| + ||v||

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Q: Let R = {(n, −n) | n ∈ Z and n ≤ 0} ∪ {(n, n + 1) | n ∈ N}. Is R a function? Justify your answer.

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Q: B. Can the proposition (Vx, y, P (x = y) → (P(x) → P(y))) be proven using the equality axioms?

A: Look at the following equality axiom. ∀x,y, and any function f,x=y⇒fx=fy

Q: 1. Find the absolute extreme points {x-cord., y-cord.), max the given interval, f(x)=x' - 3x² – 4…

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Q: Find ||F||. F(x, y, z) = i + j + k

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Q: JJ Let Prove that V₁, V₂ and V₂ E IR || V₁ + V₂ +vall ≤ ||v₁|| + ||v₂|| + ||V3|1

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Q: Prove: If f1, f2 € R(a) on [a, b] with fi < f2, then ' fi da < s' fz da

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Q: Prove f:N - N. F(X) = 5x is injoctive.

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Q: B. Let S and T be sets of real numbers, and define S. T to be {x.y:x e Sandy € T}. Show that if S…

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Q: 2.59 Find || f |sup if a) f(x) = x on (-1, ). So if x E Q, b) f(x) = – x² .2 if x ¢ Q. -

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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: 1. Translate these statements into English, where the do- main for each variable consists of all…

A: a) for all x, there exists y such that x is less than y b) for all x, for all y if (x is greater…

Q: 1. Let a be a positive real number and n be a natural number. Define S = {x €R: x >0 and x" a?…

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Q: Prove that ||A||1 = max1<j<n{£", |aij|}. %3D

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Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

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Q: (B) Prove that (Q,|·|) is not complete.

A: A metric space (X,d) is said to be complete iff every Cauchy sequence in X converges to a point in…

Q: 5. Prove that if ƒ : [-1, 1] → R is measurable, then |f| is also measurable.

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Q: Theorem 4 If B CA, P(B)< P(A).

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Q: 1. Let a, b, c be positive real numbers, and let p, q, r be [0,1/2]. Suppose a+b+c= p + q + r = 1.…

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Q: 7. ) Let f: N→ Z with f(n)=n+4. Let A = {1,2,3} and B = {1 € Z \ I <10). Use these to find the…

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Q: 15 Let (x+1) {x-2}" 20 g"k) = concave upX-) on what interval(s IS (NOte: Crinen gil but aske about…

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