Exercise 3.28. It is obvious that if T1 is both coarser and finer than T2 on X, then the topologies are equal (we say the topologies on X are equivalent). Prove it anyway. Then give a reason why we might have chosen the "coarser/finer" language.

Q: 4. Show that if ki = (X; – X)/[Σ(X; – Χ)2], then (a) Σ ki = 0; = (b) Σk? = 1/[Σ(X - X)]; (c) Σk;(X;…

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Q: For a E N, a > 1, is a + 1 always invertible, always a zero divisor or sometimes one or the other in…

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Q: A signal is given as x[n] = 8[n] + 3 & [n – 1] + 5 8[n – 2] + 8[n – 3] and impulse respond h[n] = 2…

A: Solution

Q: Prove that S4 is not isomorphic to D12.

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Q: The order of the element 3 in Z24 is

A: Given: Z24 Formula Used: The order of an element k in Zn is given by Ok=nhcfk,n

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: Prove for all mEw and n E w, we have that m+ ·n=m・n+n.

A: Given that m,n∈ω. To prove: m+· n=m·n+n

Q: The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|,  or rather the…

A: Introduction: If p is a prime number and p|n, then ϕ(pn)=pϕ(n). Proof. You'll see that every number…

Q: Prove that gcd(m + 1, n + 1)|mn - 1 for all m, n E Z \ {-1}.

A: Let, gcd(m+1, n+1)=x From the properties of GCD, we can say that, x divides m+1 and n+1 Now,…

Q: Is it true that if a, < b, for all nE N and E-1 a, is divergent, then E-1 bn is divergent? If false…

A: Given: an≤bn for all n∈ℕ To check that if ∑n=1∞an is divergent then ∑n=1∞bn is divergent

Q: The characteristic of Z5[x]/ is O 3 5

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Q: Find the elements (a, b) of G of order 2. O Show that G has no element of order 3.

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Q: Another means of rejecting Ho is if the p-value is greater than a. O less than a. O less than 0. O…

A: We have to find the another means of rejecting null hypothesis is the the p-value is ?.

Q: Find the least upper bound and the greatest lower bound of S = { 2pg+1 PEN, q E N pq and prove that…

A: The given set is  S = {2pq+1pq, p∈ℕ, q∈ℕ}. Now, we know that for any p∈ℕ, q∈ℕ.    pq &lt; 2pq⇒pq…

Q: the textbook. Then: A10 (i) Reading Questions: 1. Why we cannot say that r = Vr²? Give an example…

A: The given equation is x=x2 also the modulus of x is x=x2 then negative x could also be taken as…

Q: 3. Find a counterexample for the following: a] bd(sUT)- (bds)U (hdT)

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Q: For n>2, let an=5n/n2-2 determine for which n, |an|< 1

A: For n&gt;2, let an=5n/n2-2 determine for which n, |an|&lt; 1

Q: 10. a.If f(t) = tm and g(t) = tn, where m and n are positive integers, show that…

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Q: 2. Prove that, for all k E N, 2 can be written as the sum of two squares.

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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: 1The land of Zoo Nearland consists of two islands, the Each year 35% of the populatiou of the East…

A: (a) given let ek and wk be the population of the east and west islands respectively to find  a…

Q: 3. Let n E Z. Give a direct proof that there exist x, y E Z such that n = 55x + 78y.

A: Objective : To prove for any n∈ℤ ∃x,y∈ℤ such that n=55x+78y Prove for n=1, Greatest common divisor…

Q: 2. Reduce the following second order PDES to canonical form. -(1+y2u= 0

A: The given differential equation,

Q: Find the elements of Z24 of order 8. (Hint: o(1)=24)

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Q: . Give the Dedekind cuts in R 20 corresponding to the following. Your definition should not refer to…

A: Given: 10 23 23+3 To find: Dedekind cuts in ℝ≥0 for the given real numbers.

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

A:

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: m Show that E (")C")= ("*"). Using Combinatorial Proof. \m+p. k=0

A: We will use equating coefficent method

Q: Suppose E(X)=2, E(Y)=4, E(Z)=10. X, Y, Z are dependent. What is E(2X+3Y+Z)? Can not compute with…

A: Given,E(X)=2E(Y)=4E(Z)=10X,Y ,Z are dependent

Q: (a) Prove the following property: (:) - (.".)

A: Hey, since there are more than two questions posted, we will answer the first question. If you want…

Q: Use an indirect proof to complete the following proof. M. Given: AMNQ with N mZN = 120° Prove: ZM…

A: Consider the provided information, ∠N=120° We know that, If sum of two angle is equal to 90 degree…

Q: The characteristic of Zs[x]/ is 6. O None of the choices

A:

Q: Let dg denote the number of derangements of order k. (a) Prove the identity C) d = n! with a…

A: Given dk is number of derangement of order k. We know that for a set 1,2,…,k, the derangement dk are…

Q: Bb Announcements - Spring 2020-C X W a-concrete-introduction-to-highe X b My Questions | bartleby…

A: Let

Q: The Euler totient function is defined as ϕ (m) = |{k : 1 ≤ k ≤ m gcd(k, m) = 1}|,  or rather the…

A:

Q: If m, n are relatively prime, then o(mn) = o(m)o(n)

A:

Q: When A has full rank, the pseudo-inver en m≥n, and as A+ = AT (AAT)-¹ when n A = I, so A+ is a left…

A:

Q: 4. Show that if ki = (X; - X)/[Σ(X; – X)2], then (a) Σki = 0; = (b) Σk? = 1/[Σ(X; − x)2]; (c) Σκ;(X;…

A:

Q: 2. Suppose that {v1,... , Vp} is an orthonormal set in R", and that x E R". What can you say about…

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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: Estimate the value of f2 (1 – x-4x + 2x) da using 4- segment Simpson's 1/3 Rule. O 1144.5 O 2146.7 O…

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Q: Theorem 26. If n ∈ N then (1^n) = 1. Theorem 27. If x, n, m ∈ N then (x^m)^n = x^(mn) . Exercise…

A: Given 1n =1 where n belongs to N(natural numbers)   Prove .. 1×1×1×1.........×1 n Times.........(1)…

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: 3.1 For Aand B subsets of the univers al se tx iprovide a formal prost for: A x CBu C) is a subset…

A: Let A, B  and C are the subsets of the Universal set.   We have to show : A×(B ∪ C)  is a subset of…

Q: ***PLEASE HELP WITH THE FOLLOWING AND PLEASE PROVIDE HANDWRITTEN WORK ON PAPER*** (4) Find the…

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Q: 8. Working over R, suppose that M = PDP ' where P 1 Then, for any positive integer k, we have that…

A: To find Mk =?

Q: Exercise 5.3.12. Suppose a E G with |a| = n < ∞ and there is k e Z such that ak multiple of n. = e.…

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Q: Suppose that in the forward difference formula of order O(h) for f'(xo), h = 0.2 and |f"(c)| < 0.5,…

A: hf''(x)/2 ≤0.2×0.5/2 ≤0.05

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.