Prove that TIN) F(a – p) O< Re

Q: Prove that for all a € R, 0 < r² + 2 VI

A: Prove the following

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

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Q: Prove that A u (B n C) = (Au B) n (A u C).

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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: Let n > 3 where x, y , z € R", |x – y| = d > 0, and r is some positive real number. Prove that if 2r…

A: It is given that x and y are d distance apart and r&gt;d/2 where x, y, z belongs to Rn So locus z…

Q: Let x, y EZ. Prove that 3|xy if and only if 3|x x or 3 y.

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

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Q: (c) Let n be a positive integer and assume that there are integers a and b ab. Show that a < yn or b…

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Q: Let n eN 1. Show that o (n²) = ng(n) 2. Show that o (n) +

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Q: (1) Prove directly from the definition that: 3 n+1 →0.

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Q: Prove the following: P(n,r)=P(n-1,r)+r.,P(n-1,r-1)

A: Recall: P(n,r ) = Prn=n!(n-r)! n! = n(n-1)!

Q: 1. Let A=(a)xn be symmetric and positive definite. Show that a) a>0, i=1,2,...,n. |a|< (aa)², i,…

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Q: Is 1z = 1 for all z ∈ C?

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Q: Prove if a e G, o(a) =n, then a = reN GCD(n,r)'

A: Given below the detailed solution

Q: If a € R such that 0 0, then show that a = 0.

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Q: ). Find 1 |z-j|=2 Z

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Q: v. Let Ac S, prove that P(Ā) = 1- P(A)

A: Given that

Q: Let x € Z. Prove that if 2 | (x²-5) then 4 | (x²-5).

A: Given that .The objective, if then prove that .

Q: Proue that || A|1maπi <j <n 1 Σajl . m

A: In the given question we have to prove that   A1 = max1≤j≤n{∑i=1maij}                         (1)the…

Q: If A belongs to GL(2, R) and |A| is finite, what can you say about det A?

A: Given that A belongs to GL(2, R) and |A| is finite We know that GL(2, R) is collection of all 2×2…

Q: let n be a positive integer. prove that (X nY) nZ = X n(Y nZ) for all x,y,z E Z

A: You can prove the statement that (X∩Y)∩Z=X∩(Y∩Z) for all sets X, Y, and Z by using the distributive…

Q: Prove that [d = nt(n)/2 dẫn

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Q: Prove for a e Z and n e N that gcd(a, a + n) | n.

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Q: Prove that Σ(;) = 0.

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Q: Let x, y € Z. Prove that if 2 | (x² - y2) then 4 | (x² - y²)

A: We know that the sum of two odd numbers is an even number.Also, the difference between two odd…