(5a) Let A = {3, 4} be a subset of S = {1,2,... ,6} and let n E S. || n2(n+1)² Prove that if is even, thenn E A. 4

Q: Prove that if a, bER and a < b, then |(a, b)| = |[a, b)| = |(a,b]| = b – a. %3D %3D %3D

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Q: 5.2.6 Prove by induction that, for all n E N, 1 1.2+2.3+3.4+ · .+n(n+1) = ;n(n+1)(n+2) ...

A: Given statement is 1.2+2.3+3.4+...+n(n+1)=13n(n+1)(n+2). Induction proof is done in following steps.…

Q: 1. Let n be a positive integer and le {1,...,n} such that n divides a, (ai+1-1) for i =

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Q: 26. Let E be the set of all even integers, and let O be the set of all odd integers. Let X = = {n =…

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Q: 14. Let (a, b) = 1 and 2 = y for some integers x and y. Then prove that there exists an integer n…

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Q: #3

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Q: This exercise uses one or more of the following results from Section 5.2. Exercise 5.2.10: For every…

A: We will use the given information to complete the incomplete proof of given statement.

Q: 8. Let = {(0,²) In = {(0,²) n = 1,2,3,...}. Prove that n = 0.

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Q: Prove that for every n E Z, n² – n is even. (Proof by cases)

A: In the given question we have to prove that for every n∈ℤ, n2-n is even.

Q: Consider the set A={ Identify An B. 3n+2 4n-1 4n+1 {In € N*}. 5n-3 In € N*} and B={

A: Kindly refer to Step 2 for the solution.

Q: 3.5. Let n E N. Prove that if n + ÷ < 2, then n² + n2 <4.

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Q: 1. Let a be a positive real number and n be a natural number. Define S = {x E R: x >0 and " a?…

A: Note: According to our guideline we will answer the first three subparts of the question as the…

Q: 13. Use induction on k to EIR", then X₁1 | x,t... Xklls x + prove that if +||XK||

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Q: Let X₁, X₂,..., X be a r.s. from N(,0) and Y₁, Y₂...., be another independent r.s. from m…

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Q: Prove that for all n € N, 91(4" + 6n + 8) and n € N, 31(n³-n).

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Q: * 30. Let n and k be integers with 1 ≤ k ≤n. Show that n Σ() ( k=1 k n - 1) = (2n+2)/2 - (2n).

A: The answer for the given questions has been solve step by step below please find the attachments…

Q: 2.a) Ση-0(- 1)" χ" = 1+x for |x| < 1

A: This is a question of real analysis. L.H.S is a geometric series with common ratio (-x)

Q: Verify the following relations: a) n=1 1 Σ(n-1)³ n=2

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Q: Consider the numbers sn defined by sn = 1 + 3 + 5 + 7 + … + (2 * n + 1), for all n ∈ ℕ. That is, sn…

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Q: Prove that if m | k then 2" – 1| 2k – 1

A: Use the definition of Divisibility, "a divides b" means a and b are integers and there is an integer…

Q: The function f: {0; 1}³ → {0; 1}defined as follows: f(x,; x,;X,)=(3 x,+ x +2 x;) mod 2, then f(1; 1;…

A: We are given a function  f: {0;1}3 → {0:1} defined as:f(x1, x2, x3) = (3x1+x2+2x3) mod 2To find:…

Q: 15. Let z = 2cos0 + i(1 – 2sin0) where -n < 0 < n Show that Iz – il = 2

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Q: #4

A: It is needed to prove that:

Q: Prove that n < 2n for all n ∈N

A: We prove n&lt;2n by principle of mathematical induction Principle of mathematical inductionStep 1:…

Q: 1. Prove that 12 + 22 + ...+ n² : n(n + 1)(2n + 1) %3D 6. for n e N.

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Q: Prove that µ(d) p(n) = n . P ulp

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Q: Let X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently. 1.…

A: X1 ∼ N(0.7,1.4), X2 ∼ N(−0.5,1.2), X3 ∼ N(0.9,1.7), and X4 ∼ N(−0.2,2.1), independently.

Q: all integers h, k, and with

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Q: 2. Let ECR. Show that E is Lebesgue measurable if and only if there exists a sequence of open sets…

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Q: n Exercise 2.11. Prove that n = [2] + 10 for all n E Z.

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Q: (1.2.7)List all elements in the following sets. (a) T = {m € Z | m = 1 + (−1)¹ for some integer i}…

A: The sets are: (a) T=m∈Z | m=1+-1i for some integer i (b) U=r∈Z | 2≤r≤-2 (c) U=r∈Z | -2≤r≤2 To Do:…

Q: Remark: For positive integers n, m, we have • (n) n (m) (n) + (m) = = (lcm (n, m)) (gcd (n, m))

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Q: η Σ() = 2 – n+2 2n

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Q: Suppose n1,n2 are positive integers and gcd(n1,n2) = 1. Then for every integer x, if n1 | x and n2 |…

A: Given that gcd(n1,n2)=1 Then there exist integers m1 and m2 such that          m1n1+m2n2=1 Then…

Q: 4. Prove that if (an) is increasing and an ̸= 0 for all n ∈ N then ? 1/ an is decreasing.

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Q: Let n  ∈ Z with n>0. Prove that n | (n+1)n  - 1.

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Q: Let X-N(65,36) P(x<70)= O 0.50798 none O 0.98077 O 0.76973

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