Suppose a E Z. Prove that for any n E N, 5|2"a implies 5|a.

Q: Let S, := (1 - (-1)"/n :n e N). Find inf S, and sup S,.

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Q: For any n e N, let An = {x € R | –n <x < n}. Prove that NneN An = A1. (Recall that we define 0 ¢ N.)

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

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Q: 4. Prove that for n > 0: E, 2* = 2"+1 – 1. sk3D0

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Q: For each n e N, let A, = (}, } + ) %3D Find Unen An. Find Nnen An.

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Q: 0 <m < n such that am = e. (Hin

A: Subgroup generated by any element in a finite group will have at most n element where n is order of…

Q: Suppose that > ak = a and let c E R. Prove, in all the that (c-an) = gory detail necessary, = c•Q.…

A: We will give two proofs one based on partial sums and the other will be a direct proof.

Q: 7. If [G: H] = 2, prove that gH = Hg (18 from 6.5)

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Q: 1) Suppose A e M„(R) is. show: A) A is singular if and only if adj (A) is singular. n-1 B) If n > 1,…

A: The solution is given as

Q: 1. Given: m |l n, 2l = 23 Prove: a | b 21 3 Proof:

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Q: 3. Prove the Archimedean Property, if x ER. then there exists K E N such that r SK.

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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: 4. Work either (a) or (b), NOT BOTH (a) Let In = for n E N. Prove that nIn = 0. 0, n n%=D1

A: a. Given that    In=0,1n  for  n∈ℕ. And we have to prove that…

Q: For any x ER, prove that there exists q E Z such that q-1<r <q.

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Q: :""n e Z*, if n³ = 3k for some k E Z, then n = 3q for some q E Z." %| %3D

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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: 3.19 Suppose if r = 0, f(x) = {-1)" if r E n+1' n where n E N. Prove: f € R[0, 1], even though f has…

A: Sol

Q: Suppose M 4 G and M < G. 1) Prove that Mn N 4 G. 2). If M N N = {1}, show that for m e M, n E N, mn…

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Q: If E, and E2 are measurable then (E,UE) is not measurable .

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Q: Let m eN and a, b e Z be such that (a, m) = 1 and (b, m) Let h = 0m(a), k = 0m(b), and let n = 0m…

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Q: a n eln t--+

A: Given a function : h(x)=31-2x+2 We need to find the graph of this function.

Q: For positive integers n and k (k sn+ 1), the kth term of (a + b)n is given by

A: The binomial expansion of (a+b)^n is

Q: For all n E N, let An = {0, 2n, 4n, 6n, . . {x € Z|(3n E NU{0})(x = 2n)}. (a) Find (with proof) [U…

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Q: Show that E [(X-µ)^n]=0 for ∀n, with X~N(µ, σ²) and n≥3 being odd numbers.

A: We need to show; E[(X-μ)n]=0 for ∀n and n≥3 being odd numberswhere;X~Normal(μ,σ2) That is, we need…

Q: If e'z = /3 -i.Then Im (z) is %3D O a. e2 b. 0 С. 1 d. -In(2)

A: eiz=3-i Now let z=x+iy. Then, ⇒eix+iy=3-i⇒eix+i2y=3-i⇒eix-y=3-i…

Q: Show that for all n E Z, n² 5.

A: Consider the given condition: This shows that whatever will be the values of n, it will always be…

Q: Show that for all n e Z, n? – 3n +17 2. -

A: Given: For all n∈ℤ, n2-3n+1≠2

Q: Compute the following: remember to show your working. (a) N An if An = (1 – ,n] + (e.g. x E A3 →} <…

A: We will partition An and then find the intersection of all An as n tends to infinity.

Q: Let p denote 3x E Z such that P(x)" and q denote dx E R such that P(x)". Ğive a simple definition…

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Q: (5) Show that T(n)o(n) > n for all n E N. 11

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Q: Show that E(SSE) = (n - k-1), as in Theorem 7.3f, using the following approach. Show that SSE = y'y…

A: Please find the explanation below. Thank you

Q: 28. (a) Prove that for x e R" and s < t, Ds(x) C D;(x).

A: The notation Dsx represent  an open ball of radius s, centered at x∈ℝn. Let x=x1, x2, …, xn,…

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: If E (ar + br) 2n and n+1 k=1 Σ (ακ-bk) -- 1, then Σa- n+2 k=1 k-1 O 2 3 2 O 1 5. 2 Ο None

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Q: Q: Prove that : sin' ne -de = na for all n e N. sin' e

A: solution is given below

Q: H w/ Prove the lemma. if E(IXi) <o then for rarE (IX)c

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Q: If m` E = 0, then E is measurable.

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Q: 1. For an = 5n2 + 9n3 + 1, an = n2, prove or disprove: rn = O(an) (n → o)?

A: We will use the standard definition.

Q: 3. ncnt) use PMI to prove that E

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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: et A E C. Show that |A| 1 if and only if A= e for some 0 E R. %3D

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Q: What are the value for sup e-* = е x>0 and inf e * - = XER

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Q: For any n e N, let Bn = (n, ∞0). {B„}1 is descending. Show that Bn = 4. %3D n=1

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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: If det A= 0 why is det A+ = O? If A has rank r, why does A+ have rank r?

A: Given Data det A=0   Suppose the column and row of the matrix are linearly independent. The matrix…

Q: (a) Prove that x, y = -1, y = 1 1 (b) Prove tha and n: m and

A: This is a problem of Number Theory.

Q: Let x, y,p E Z, where p is prime. Prove that x2 – p²y – p + 0. (Hint: you may use without proof that…

A: Solution :-

Q: D. Is there guaranteed to exist a value of x = c on -2 < c < 2, such that h'(c) = 0? Give a reason…

A: Rolle's theorem states that if a function f is continuous on the closed interval [a, b] and…

Q: 4) Let x, y, z E Zg. Prove or disprove the following: Ifxy = xz, then y = z.

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Q: Let f(n) = 2n – n? + 10n – 7. Show that f(n) is O(n³) using specific values of C and ng-

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Matrices

‘Matrices’ is the plural form of the word matrix, and it is basically a spreadsheet in the form of a box. In mathematics, various functions can be carried out with matrices. Generally, a matrix comes in the shape of a square or rectangle. The elements are distributed horizontally in the form of rows, and vertically in the form of columns.