4. Let R be the set of all real numbers, xER and n EN such that n > 2. If (1+x) >0, prove that Pn : (1+x)" >1+nx is true V n E N.

Q: 1. Let y be a positive real number. Prove that for every n e N there exists a unique positive real…

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Q: 1. Let ƒ : N × N → N be defined as f(m, n) = 2m−¹(2n — 1). (a) Show that f is a bijection. (b) Using…

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Q: Find the value of k such that A is singular. 1 0 5 3 -1 0 2 k k =

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Q: Let X = {x E Z|x = Y = {y € Z]y = 3n for some integer n}. 6m for some integer m} and let %D a. Prove…

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Q: Find the value of k such that A is singular. 1 0 4 5 -1 0 3 3 k k =

A: Topic:- Matrix

Q: Let x, y e Z. Prove that if 3x+7y is odd or 5x + 6y is even, then x or y is even.

A: We have given that for x,y belongs to integers. If 3x+7y is odd or 5x+6y is even then x or y is even…

Q: 13. Prove by Math Induction ex > 1+x for all positive integers x > 1

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Q: 6) The Ying of integers(2) is ideal of veal number (R) an DTrue D Fulse

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Q: Let x, y E R. Prove that if 21x + 35y = 45, then x and y are not both integers.

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Q: Prove using the IVT. For all positive integers k, cos x = xk has a solution.

A: Intermediate value theorem: Let f be a function continuous on a closed interval a,b. Let k be number…

Q: (b) Prove that xe" 2 for some x in (0, 1). %3D

A: By the intermediate value property of continuous function if a function f:(a, b) --> R be a…

Q: Let n > 1 be a positive integer and let a1, a2, ... , a, be real numbers. a) Prove that Σ α la1| -…

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Q: Find the value of k such that A is singular. 1 0 3 4 -1 0 2 4 k k =

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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>d/2 where x, y, z belongs to Rn So locus z…

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

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Q: Suppose that {$1,...Øn} F p and I F ø; for all i < n. Show that I F þ.

A: Proof: Suppose ϕ∈Γ∗ϕ∈Γ∗. As Γ∗Γ∗ is maximally consistent it clearly follows that there exists a…

Q: 6. Suppose that f(x) = x" + an-1r*-1+ ...+ ao E Z[x]. Ifris rational and x –r divides f(x), prove…

A: We will use division algorithm to prove the fact.

Q: Prove that for all positive integers n.

A: To prove for all positive integer n ∫01(lnx)ndx=(-1)nn!

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: Show that the function f(z) = from [0, 0) to [0, ∞) has a fixed point c. Hint: Set f(r) = z and show…

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Q: 14. Let (a, b) = 1 and r = y® for some integers r and y. Then prove that there exists an %3D integer…

A: As per policy, we are solving only the first Question please repost 2nd Question to get its…

Q: (2) Suppose that G, C Zn-r. Then prove that G,-1C Zn-r+1

A: We prove that

Q: Show that if x ~ Nm (u, 2), where N is positive definite, then (x – u)'N-I (x – µ) ~ Xm: 2

A: Given  x~Nmμ,Ω, where Ω is positive definite.

Q: prove for any 2 non negative integers x and y, where x>y, if their gcd IS Y, THEN THEIR LCM is x

A: The objective is to show that LCM(x,y)=x, given GCD(x,y)=y; where x and y are non-negative integers…

Q: 5. Prove that if x, y ER, then x + y ≤ x + y. Use the following cases: (i.) x, y ≥ 0, (ii.) x, y <…

A: We now prove that if x,y ∈ ℝ , then x+ y≤ x+y with the help of given three cases.

Q: Suppose I is an open interval and f € C°(I). Show by induction that ffor х, а € 1, f(x) = f(a) +…

A: Given: I is an open interval and f∈C∞I   To show: fx=fa+∑k=1nfkak!x-ak+1n!∫axx-tnfn+1tdt for x,a∈I…

Q: 13 3. Use the IVT to prove that y = r Inr has at least one zero on 2 2

A: Above function is continuous function in given interval. So we can apply IVT

Q: Find all the values of x in the interval [0,2pi], that satisfy the question.  10cos(x)+5sin(2x)=0

A: Given equation is Solve the given equation for x

Q: Prove that there cannot exist two different polynomials q,r E K, both of degree less than n. such…

A: Given K be any field and a1,a2,...,an be pairwise distinct elements of K. For each i=1,2,3,...,n, pi…

Q: 1. ( Prove that for any x, y e R, we have ||| - ||y|| < |× -y|l. 2. Let u e R. Prove that 0 u 0,…

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Q: If x > 0, prove that T - In (1+ x) > 2+2x

A: Using derivative

Q: Find the smallest integer in the given set a.{ xϵ Z f x &gt; 0 and x = 4s + 6t for some s, t in Z }

A: Consider the equation: x=4s+6t……1   where: x, s, t∈ℤ and x&gt;0

Q: 5. Find all pairs (m, n) of positive integers such that y(y(nm)) = n, where p is Euler's totient…

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Q: Let m and n be positive integers with gcd(m,n) = 1. Prove that Z/mZ x _Z/ nZ = Z/mnZ

A: Suppose m and n are relatively prime.

Q: Find the absolute maximum and the absolute minimum of U(q) 20002 (15 5q)4 on the nterval 04.

A: First find the derivative of the given function and compute the critical points.

Q: Determine all the number(s) c which satisfy the conclusion of Rolles Theorem for

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Q: 2 (x) 2. prove that s

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

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Q: Prove or disprove: If xn ≥ 0 for all n ∈ N and lim sup xn = 0, then xn → 0

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Q: Suppose that f(x) = x" + ax"-1 + . . . + a, E Z[x]. If r is ra- tional and x - r divides f(x), show…

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Q: 4. Prove that d x" = nx"-1 ミクー! de

A: To prove that ddx(xn) = nxn - 1

Q: Prove that 2n+l > (n + 2) · sin(n) for all positive integers n.

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Q: Q1. Prove that KT = = -.

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Q: Find the value of k such that A is singular. 0 4 6 -1 0 4 2 k k =

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Q: 4. Verify thatf (x) = x³ – x2 – 20x + 6 satisfies the three conditions of Rolle's Theorem on [0,5].…

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Q: Let a E Z, a #0. Prove that for any positive integers m, n with n<m, a2" +1 divides a2 -1. a) a even…

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Q: Let x +1 be a real number. Prove that for all n E N, x" – 1 - + x" +...+ x2 + x + 1. x – 1

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Q: Let x, y e Z. Prove that if 3x + 7y is odd or 5x + 6y is even, then x or y is even.

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.