Define X Σ=1 Χ n Show the following using the properties of summation: a. nẴ =). X i=1 72 b. [(x₁ - x)²=X²-nX² i=1 i=1

Q: Compute the following. 4! 6! ]*

A: Answer - First we need to understand what is factorial before solving this question  n!…

Q: (c) lim sup (an + bn) lim inf an + lim inf bn

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Q: Prove the following fact: A" = 3"-1A. 1 1 1 1 1 1 1 1 1 A =

A: Given, A =111111111  We have to  prove An = 3n-1A

Q: . If A and B are sets in a universal set U, then AUB=AnB.

A: We proved this by wave diagram

Q: (BOC)=(A-B)u(A-C). 11. If A and B are sets in a universal set U, then AUB=AnB. 12. If A, B and C are…

A: Please refer to the solution in next step.

Q: In this part you will show that if f(x) ≤ g(x) for all x = [a, b], then [* f(x) dx ≤ [*9(x) dx by…

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Q: Prove that {9":ne Q} = {3":neQ}.

A: Two sets A and B are equal i.e. A=B if and only if

Q: QUESTION 10 State whether the following are true or false. If true, you must prove it and if false…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: . If A and B are sets in a universal set U, then AUB=AnB. A-(BnC)=(A-B)U(A-C). . If A, B and C are…

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Q: Prove that for every integer n > 2, P(n+1, 3) = n³ . %3D -n.

A: Given to prove that for every integer n≥2,  Pn+1, 3=n3-n. We shall prove this by the method of…

Q: Question 2. The subgroup generated by S could have been defined a second way, as the set of all…

A: The subgroup generated by a set in a group , where represents the subgroup generated by and is…

Q: Compute the following. (a) P6, 2- (b) C6, 21 (c) Pa, s (d) Cg 9

A: The formula for Pn, r Pn, r  =n!n-r! The formula for Cn, r Cn, r  =n!r!×n-r! Here,  n!…

Q: Proposition 4.1.2. If ƒ : A → Rn satisfies the lower-Lipschitz condition on E, i.e., if there exists…

A: We will prove all by using the definitions

Q: (a) Let n be any integer. Prove that n2 is congruent to 0, 1 or 4 modulo 8.

A: Note: According to bartleby guidelines we have to answer only first question please upload the…

Q: All the eigenvalues of a symmetric matrix A are positive. Which of the following statements is…

A: Solution

Q: Provide a left-hand-side, right-hand-side proof of the following: dt

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Q: 12-4 If M and N are similar matrices (that is, N = some nonsingular matrix P), then oe(N) = oe(M). =…

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Q: 15. If A, B and C are sets, then (AnB)-C=(A-C)n(B-C). m

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Q: Let p(x) and q(x) be predicates with variable x. Prove the following. • (3x)p(x) (x)~p(x) I(p q) =…

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Q: I need to prove each of the following, showing all steps.

A: Prove the following (a) Let A and C are symmetric matrices, then . (b)

Q: C. The Fibonacci numbers (Fn)n21 are defined by the recurrence F₁ = 1, F2 = 1, Fn = Fn-1 + Fn-2 for…

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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: Let S= {n | n=5 k+ 4 for some integer k} Let T= {n|n=5 j– 1 for some integer j}

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Q: 29. For any odd integer n, n° n² +3 %3D 4 4

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Q: Let a 20 be a real number (to be chosen below) and recursively define an-1 + a2-1 an = 3 for n > 2.…

A: Given that a1≥0 and recursively defined an=an−1+an−133 for n≥2.Note that the given sequence is…

Q: Prove the shown?

A: we have been given limh→0sin hh=1 and limh→0cos h-1h=0 we have to use the limits to find the…

Q: (BOC)=(A-B)u(A-C). 11. If A and B are sets in a universal set U, then AUB=AnB. 12. If A, B and C are…

A: To prove the equality A - (B ∩ C) = (A - B) ∪ (A - C), we need to show that each set is a subset of…

Q: Compute the following values, showing your work: φ(20), φ(25), φ(54) and φ(8!)

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Q: 2. (a) Use the definition of the derivative and the Binomial Theorem to = nx"-1 for any x" is f'(x)…

A: Since you have asked multiple question, we can solve first question for you. If you want other…

Q: 17. If A,B and C are sets, then Ax (BnC) = (A x B) n(A x C).

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Q: 18. If A,B and C are sets, then Ax (B-C)= (A x B)-(AXC).

A: By the using of definition we prove this.

Q: B=(2,).Then 2 E3 %3D

A: Some definitions: Interval:  Let S⊂ℝ, then we say S is an interval if c,d∈S such that c≠d. Then, ∀…

Q: 14. If A,B and C are sets, then (AUB)-C=(A-C)u(B-C).

A: To prove the equality (A∪B)-C = (A-C)∪(B-C), we need to show that any element x belongs to the…

Q: Use mathematical induction to prove that for each natural number n, 3 di- vides n³ +23n. Compare…

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Q: Find the expected value of each probability mass function below. 1 3 P(x) 0.13 0.44 0.34 0.09 Select…

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Q: An integer x is odd if x2 is odd. Let x, y and z be integers. If x y and y z, then x | z. For any…

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