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: (a, o) is not measurable.

A: to find that a,∞ is measurable or not  proceed as follows We know that every interval is measurable.…

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

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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: 7 (:) Prove that f(x) : is convex.

A: Given the function f:ℝ→ℝ2 by, f(x)=xs Note that the domain of the function f(x) is convex.

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

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Q: Find all possible values of a e and Iz+2a-(a + 1)i> 3 simult

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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: For r> 0 let A set A. = w:w= exp (¹) where 0 < |z| <r} ; determine the

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

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Q: Prove that if 0 < a < y, then ỹ – VI < Vy– a.

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

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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: Let: A = {n € Z|0< n < 3} B = {Vn|n €Nand 1 <n < 4}

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Q: If b is dviisible by a (a|b) and b in different than 0 then |a| is less or equal than |b|

A: The given statement means the following.

Q: (c) Let z1 = 2+ 2i and let 22 =1+i. Prove that the triangle inequality |21 + z2| < |21| + |22|…

A: Left hand side is equal to, z1+z2=2+2i+1+i=3+3i=32+32=32 Right hand side is equal to,…

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: Determine b so that f(x) is continuous if O 8 x + 6 xS 6 f( x ) = D O 2 x2 + bx +10 x > 6 b =

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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: Let a, b € Z and both nonzero. Prove that gcd(a, b) x lcm(a, b) = |ab|.

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Q: 2F. n 1 ;;1=2" (b), (26)

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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: compute measuralle

A: [f&gt;t] is an inverse image of (t,infinity) That leads us to give an answer.

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: #6

A: We know that a Gamma density function is: So E[X] is:

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

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Q: If |z-8 <, and ly-히 < , then a - y < 1. 2'

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Q: Prove that c?p?-(jx² + jy²+jz³) is the invariant quantity equal to c?po?.

A: Consider the provided question,   Prove that c2ρ2-jx2+jy2+jz2 is the invariant quantity equal to…

Q: 4. Prove that, VR, -11-

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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: Prove the inequality |arctana−arctanb|≤|a−b|.

A: To prove the given inequality.

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

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