Let 0 E A C R. Show that if A is both open and closed, then A =R.

Q: Let J={a: aER}, then I is Borel algebra.

A: False Statement' Detailed Explanation

Q: If u (E) =0, then E is measurable.

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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: Prove that la + b + c| < ]a] + ]b| + |c[

A: Prove that,    a+b+c≤a+b+c

Q: Let m eN and a E Z be such that (a, m) = 1. Let i, j e N. Prove that if om(a) = ij, then om(a') = j.

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Q: Let A € M₂ (R) be given. (a) If AA¹ = 0n, prove that A = On. Does this hold if A € M₂(C)? (b)…

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Q: Let 0 E A CR. Show that if A is both open and closed, then A = R. (Hint: Let M = sup{t: [0,t] C A})…

A: We prove this by contradiction.

Q: Let X~U (0,0), then P(X > X²) is

A: Given that X~U0,θ fx=1θ-0=1θ, 0≤X≤θ

Q: let r ∈ R\ Q. prove that for all a,b,c ∈ Q, a+br=cr iff a=0 and b=c.

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Q: Let a e R" and let A = aaT. Show that xT Ax = (a" x)².

A:

Q: Let a, b, c E Z. Prove that if a b and alc, then Vk EZ we have a (b+ ck).

A: ab  &amp;  ac   Since ab, therefore b=a·m   ,  where m∈ℤ Since ac, therefore b=a·n   ,  where n∈ℤ

Q: Let X ~ N(0, 1). For any € > 0, show that P(|X| > c) ≤ ²c - ??

A: Note: Hi there! Thank you for posting the question. As you have posted multiple questions, as per…

Q: Notice that h is definec this.)

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Q: Let c E R and suppose that an → a. Show that can → ca.

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Q: Let x, y E Z. Prove that if 3 ł x and 3 ł y, then 3|(2² – v²).

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Q: b) Prove that if x <y and z<w, then x+z<y+w.

A: Given: Suppose x&lt;y and z≤w. To prove: x+z&lt;y+w.

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

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Q: Prove that if Tr(AT A) = 0, then A = O .

A: Proof:- Suppose A is a square matrix of order n, That is, A=aij For 1≤i,j≤n Now we need to prove…

Q: Let a, b e Z where both a # 0 and b# 0. Now suppose that a = aid and b= bịd for a1, b1,d € Z. Prove…

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

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Q: If A is similar to B, prove that AT is similar to BT.

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Q: Show that for every e>0 ther e exists P suuch that U (P, f) < U (f) + e

A: We will ise definition of U(f) and Infimum.

Q: Let X, Y be sets and f : X →Y. Prove f"(A\B) = f"(A) \ f¯"(B). for any A, B CY.

A: Solution:Given:          X, Y    are the Sets and                      f: X→ Y To prove:  f-1(A \…

Q: .Prove that la + b + c| < |a] + |b| + ]c]

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Q: Let T={[a,0) : aER}, then T is not Borel algebra

A: Given a statement

Q: ), P{|Zn – c| > ɛ} → 0 as n → o. Show that for any b on g, E[g(Z„)] → g(c) as n→o

A: *Answer:

Q: F(p), then prove that

A: simple use of integral transformation

Q: Show that for every e > 0 there exists P suuch that U (P, f) < U (f) + e

A: Since U(f)=inf{U(P,f)}

Q: Let BE Cnxn. If z" Bz = 0 for all z E C, then show that B = 0.

A: Given expression is

Q: Prove: If ƒ e R(a) on [a,b] and g e R(a) on [a, b] , then fg e R(a) on [a,b]

A: Given if f∈Rα on a,b and g∈Rα on a,b. We have to prove that fg∈Rα on a,b. Since, f∈Rα on a,b then f…

Q: Show that for any positive integers i and j with i > j, we have T;(x)T;(x) = [Ti+;(x) +T;-j(x)].

A: we have to show, for any positive integers i and j with i&gt;j TixTjx=12Ti+jx+Ti-jx Now, we know…

Q: Given: CE= FH CD GH G. Prove: DE = FG

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Q: Let ECR be measurable. Prove that L'(E)n L°(E) C L*(E).

A: Given E⊆ℝ be a measurable set. Let f∈L7E∩L9E. Let λ=716, therefore:…

Q: Let f, g, h : N→ R²0. Prove or disprove that if f + g € O(h), then ƒ e O(h) and g E O(h).

A: Given f,g,h:ℕ→ℝ≥0 such that f+g∈Oh. We know that f∈Oh, if h is strictly positive for all large…

Q: Suppose A · B = 0. Show that this implies that C(B) is contained in Null(A).

A: Suppose A.B = 0 then we have to show that  Column space of B contained in the null space of A  i.e…

Q: Prove that Àx (Bx)= A.¿)B CA.B) wheve. A.a,t.azj+a3k. .....

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Q: let X~~ Per(P) Prove ô is unbiare if ô-X let x exp(0) Prave e is unbiar

A: Introduction: One random variable follows Bernoulli with parameter p and other random variable…

Q: Show that if A and B are similar, then AT and BT are similar.

A: Let A and B are similar matrices. By definition of the similar matrices.  The matric A and B are…

Q: (c) If A E M,(R) is non singular and n 2 2, prove that det(adjA) = (det A)"- (d) If A E M,(R)…

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Q: Let z E C. Prove that if 1+z +...+z"-1 – nz" = 0, then |2| <1.

A: Given that, z∈C and, 1+z+...+zn-1-nzn=0 It is known that, 1+z+z2+z3+....+zn-1=1(rn-2-1)(r-1) So the…

Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: For any x E R, there exists n EN such that x < n; ylgn ihi

A: We have to state whether given statement is true or false.

Q: Show that Oa = {a} if and only if a e Z(G).

A: For a∈G, Oa denote the orbit of 'a' under the conjuction action.The orbits under the conjucation…

Q: Find conditions on a, b, c, and d such that

A: Find conditions on a, b, c, and d such that  B=abcd commutes with both 1000 and 0001 For any matrix…

Q: Suppose that for every r E R, pqr 0. Show that R is isomorphic to R/I, × R/Iq.

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Q: Let w = r. In(xyz), then which of the following is 2 O a. O b.0 O d. O e.

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Q: Let E ⊆ R. Show that E is measurable if and only if χE is measurable

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Q: Prove that (A')' = A for each A E Mmxn(F).

A: