Ask Question, Ask an Expert

+1-415-315-9853

info@mywordsolution.com

Ask Math Expert


Home >> Math

(1) (a) describe why Z X Z must be countable.

(b) By part (a) we know that N ≡ Z X Z, and hence there must be a one-to-one correspondence between N and Z X Z. Provide one such one-to-one correspondence f : N → Z X Z. Note, it may be easiest to describe your map using a sketch of Z X  Z.

(2) Proved in that a finite product of countable sets is countable. It is not true however that a countable product of count- able sets must be countable. Here you will see one ex of a countable product of  finite sets that is not countable.

For each i ≡ N let Ai = {0,1} and let

1150_Finite sets.jpg

the set of all sequences of 1's and 0's. The set A is a countable product of  finite and hence countable sets. Prove that A is not countable.
(3) A complex number z is "algebraic" if it is a root of some integral polynomial (a polynomial with integer coefficients). That is z is algebraic if there is some

p(x) = a0 + a1x + a2x2 +........+ akxk ai ≡ Z and at least one ai ≠ 0 with p(z) = 0. For ex, the polynomial p(x) = -4+4x-x2+x3 factors to p(x) = (x- 1)(x - 2i)(x + 2i) and so has roots x = 1 and x = ± 2i, thus 1, 2i and -2i are algebraic numbers. Let A denote the set of all algebraic numbers in C, and prove that A is countable by the following steps.

(a) For each n ≡ N, let Pn denote the set of integral polynomials of degree n. Prove that for each n the set Pn is countable.

(b) Now let P be the set of all integral polynomials. describe why P must be countable.

(c) Given a particular integral polynomial p(x) of degree k, let Rp be the set of all roots of p(x). What can be said about the number of elements in Rp?

(d) Using parts (a), (b) and (c), prove that A, the set of all algebraic integers, is countable.

Math, Academics

  • Category:- Math
  • Reference No.:- M91428

Have any Question? 


Related Questions in Math

The height of north american adult men varies according to

The height of North American adult men varies according to a distribution that is approximately normal with a mean of 70 inches and a standard deviation of 3.5 inches. Using the 68-95-99.7 Rule, between what two heights ...

T angle between two forces of 37 n newtons and 55 n is

The angle between two forces of 37 N (Newtons) and 55 N is 36°. Find the magnitude of the resultant force. Do not round until the final answer. Then round to the nearest integer as needed.)

In given figure is intermediate code to compute the dot

In given Figure is intermediate code to compute the dot product of two vectors A and B. Optimize this code by eliminating common subexpressions, performing reduction in strength on induction variables, and eliminating al ...

Define the number of all the arrangements of 8 objects

Define the number of all the arrangements of 8 objects choosing 2 at a time without repetition. A. Order is important. (permutations). B. Order is not important (combinations). Please show problem worked out

Two balls are drawn in succession out of a box containing 5

Two balls are drawn in succession out of a box containing 5 red and 2 white balls. The balls are drawn without replacement. What is the probability that both balls drawn are red?

A 22 -foot extension ladder leaning against a building

A 22 -foot extension ladder leaning against a building makes a 68.7 degrees angle with the ground. How far up the building does the ladder touch? What is the distance between the ground and the point at which the ladder ...

Can anyone explain me like when we use the phythgorean

Can anyone explain me like when we use the phythgorean triplet in triangle there are many triplet so how we know when to use this one.

A city has a population of 380000people suppose that each

A city has a population of 380,000people. Suppose that each year the population grows by 8%. What will the population be after 10 years?

Your goal is to describe the details of a more-or-less

Your goal is to describe the details of a more-or-less feasible business in the renewable energy or more general sustainability sector. Do not propose a major enterprise: propose something that is feasible (or close to f ...

1nbspyou deposit 6000 in an account earning 6 interest

1. You deposit $6000 in an account earning 6% interest compounded monthly. How much will you have in the account in 10 years? 2. You deposit $3000 in an account earning 8% interest compounded  continuously . How much wil ...

  • 4,153,160 Questions Asked
  • 13,132 Experts
  • 2,558,936 Questions Answered

Ask Experts for help!!

Looking for Assignment Help?

Start excelling in your Courses, Get help with Assignment

Write us your full requirement for evaluation and you will receive response within 20 minutes turnaround time.

Ask Now Help with Problems, Get a Best Answer

WalMart Identification of theory and critical discussion

Drawing on the prescribed text and/or relevant academic literature, produce a paper which discusses the nature of group

Section onea in an atwood machine suppose two objects of

SECTION ONE (a) In an Atwood Machine, suppose two objects of unequal mass are hung vertically over a frictionless

Part 1you work in hr for a company that operates a factory

Part 1: You work in HR for a company that operates a factory manufacturing fiberglass. There are several hundred empl

Details on advanced accounting paperthis paper is intended

DETAILS ON ADVANCED ACCOUNTING PAPER This paper is intended for students to apply the theoretical knowledge around ac

Create a provider database and related reports and queries

Create a provider database and related reports and queries to capture contact information for potential PC component pro