Natural numbers Definition and 141 Threads

Question 1) Write ⊆ or ⊄: {x/(x+1) : x∈N} ________ QNOTE: ⊆ means SUBSET ⊄ means NOT A SUBSET ∈ means ELEMENT N means Natural Numbers Q means Rational Numbers Question 2) Which of the following sets are infinite and uncountable? R - Q {n∈N: gcd(n,15) = 3} (-2,2) N*N {1,2,9,16,...} i.e...

$a,b,c,d,e,f,g \in N$ $a<b<c<d<e<f<g$ $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+\dfrac{1}{d}+\dfrac{1}{e}+\dfrac{1}{f}+\dfrac{1}{g}=1$ please find one possible solution of a,b,c,d,e,f,g (you should find it using mathematical analysis,and show your logic,don't use any program)

I have seen a lot of examples of sets with same cardinality as the natural numbers. For instance the even numbers or the cartesian product. In any case the proof amounted to finding a way of labeling the elements uniquely. But I am curious - can anyone give me an example of a set, where this...

Homework Statement Taken from Spivak's Calculus, Prologue Chapter, P.28 b) Notice that all numbers in Pascal's Triangle are natural numbers, use part (a) to prove by induction that ##\binom{n}{k}## is always a natural number. (Your proof by induction will be be summed up by Pascal's...

Sorry about the intriguing title; this is just a continuation of the discussion in https://driven2services.com/staging/mh/index.php?threads/5216/ from the Discrete Math forum. The original question there was how to introduce mathematical induction in a clear and convincing way. Since the current...

is there any way to prove or disprove the statement: y=3x^2+3x+1 is prime for all x belongs to natural numbers...

Given n numbers x1,x2...xn belong to N. x1+x2+x3...xn=m How many different combinations of x1,x2,x3...xn are there?Is there any formula useful here? Note:x1,x2,x3... need not be distinct and also can be 0. Thanks

EDIT: Found the answer, seems I overlooked part of the solution in the learning materials ( answer extended into another page) the Solution does indeed equal what i thought it did. Homework Statement So this is the problem i have: (2(-1)^n -((n*pi)^2(-1)^n)-2)*(8/(n*pi)^3) where n...

I'm trying to write down an axiomatic development of most of mathematics, and I'm wondering whether it's logically permissible to use natural numbers as subscripts before they have been defined in terms of the Peano Axioms. For instance... the idea of function is used in the Peano axioms...

If σ(N) is the sum of all the divisors of N and τ(N) is the number of divisors of N then what is the sum of sum of all the divisors of first N natural numbers and the sum of the number of divisors of first N natural numbers? Is there any relation between σ(N) and τ(N) functions? Can I do that...

Prove that $$a^3+b^3 \ne to \ c^3 \ if \ a,b \ and \ c \in \ {N}$$ This is not a challenge,I am asking for help... Any help is appreciated... Thanks...

i am studying real analysis from terence tao lecture notes for analysis I. http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ from what i understand , property is just like any other statement. for example P(0.5) is P(0) with the 0s replaced with 0.5 . so the notes says (assumes ?)...

Homework Statement For all non zero natural numbers n prove that: 1- 24\mid n(n+1)(n+2)(n+3)(n+4) and that : 2- 120\mid n(n+1)(n+2)(n+3)(n+4)(n+5) The Attempt at a Solution 1- For n=1 we get that 24 divides 120 so we assume that 24 divides n(n+1)(n+2)(n+3)(n+4) and we...

The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8...

The set of all possible streams of brain activity arising from all possible configurations of all possible neurons with all possible connections is finite, so if you accept that natural numbers are a creation of the human mind (brain), then don't you have to accept that the set of number is...

The set of all possible streams of brain activity arising from all possible configurations of all possible neurons with all possible connections is finite, so if you accept that natural numbers are a creation of the human mind (brain), then don't you have to accept that the set of number is...

All, Could some one tell me please ,why in natural numeric system we are not allowed to move from number one to three? I mean ,is this a properties of the natural numbers to be in sequence? for instance if we count base on an imaginary numeric system which allows us to shift between numbers...

Homework Statement Prove that the product of two consecutive natural numbers is even. 2. The attempt at a solution Hi, I'm just starting to work with proofs by induction, I'm just wondering if this is a valid technique, and/or if I am being too verbose in my proof, thanks! Let...

I know the history of how set theory came about and how Cantor showed the real numbers between (0,1) were non-denumerable. He did this by showing that they can't be put into a one-one correspondence with N (1, 2, 3...) ...So what does that really tell me? I know it tells me that the...

Hi everyone. I have learned that: 1+2+3+...=\frac{n(n+1)}{2} 12+22+32=\frac{n(n+1)(2n+1)}{6} I want to know what the general formula of Ʃna, in which n and a are natural numbers, respect to n and a.

I have array of natural numbers from 1 to n. They are divided into m groups, where m*(m-1)=n. I need all m-1 elements from first group, last m-2 elements from second group, last m-3 elements from third group...zero elements from last group. For example 5*4=20: 1,2,3,4; 5,6,7,8...

Problem: let $\mathbb{N} = \left\{0, ~ 1, ...\right\}$ be the set of natural numbers. Prove that $(\mathbb{N},~\le)$ is a poset under the ordinary order. Solution: let $x \in\mathbb{N}$, then $x \le x$ as of course $x = x$. If also $y \in\mathbb{N}$, then $x \le y$ and $y \le x$ implies $x =...

Are there an \aleph_0 # of natural numbers with an \aleph_0 # of digits?

Homework Statement I have two problems that I got stuck. 1. \exists ! N\in P(X) , A\Lambda N=A, \forall A\in P(X) and for each A\in P(X), \exists ! A'\in P(X) , A\Lambda A' =N 2. Prove a+(b+c) = (a+b) +c, for positive integers a, b, cHomework Equations 1. Given sets A,B \in P(X), where P(X)...

Is the set of natural numbers the only infinite set that is not a power set of another set?

Hello all. I have been reading Halmos's Naive set theory. In chapter 12., there is an excercise problem which states I thought a lot about this but this seems like a theorem to me and is not at all trivial to prove it. I would greatly appreciate it if you can give me a hint or a clue as...

I wish to show that the sum of 9 consecutive natural numbers: n(n+1)(n+2)(n+3)...(n+8) will not be a perfect square. This problem came by as a result of another problem I was doing and I'm wondering if anyone knows/has come across this already. After some searching I found that n! is not a...

Is it possible to define an abelian group on the natural numbers (including 0)? It's just that for every binary operation I've tried, I can't find an inverse!

If 'n' is a natural number such that n>1, prove that there exists a natural number k such that n-k=1. It raises the question: What is the definition of a natural number? Could you say that because n is natural, n-1 must be natural so that if k = n-1, n-k = 1? BiP

I'm trying to prove that there are no natural numbers x and y that satisfy the equation x^2 - y^2 = 2. I tried to solve it by contradiction and so I assume that x and y are rational numbers and both x and y can be written in the form (a/b) where it's in its simplest form and a and b are both...

My teacher was saying that we can't have a set of infinitely decreasing natural numbers. What if we started at ω and then worked our way backwards. I realize that is ill defined. And where ever we start will be a finite number. But if we can have an infinitely increasing set in the...

Greetings, comrades! In a previous thread, a user articulated a common argument: His analogy mapping knights to horses makes intuitive sense, but how can we apply this idea to two infinite sets of knights and horses? How can we treat finite and transfinite sets equal in that sense and...

Homework Statement Show that A, the set of all increasing sequences of natural numbers is uncountable Homework Equations I know that the natural numbers themselves are countable. The Attempt at a Solution I am thinking of using some sort of diagonal argument to prove this.

Homework Statement Prove by Mathematical Induction that the assertion, n ∑ r^2 = n/6 (n+1)(2n+1) r=1 holds for every natural number n. Homework Equations Ok, so basically, how do you solve this question? I have got to the Induction step but I'm not sure how to do it...

Homework Statement N refers to the set of all natural numbers. Part 2: From the previous problem, we have σn : N → N for all n ε N. Show that for any n ε N, σ(n+1)(N) is a subset of σn(N), where we have used n + 1 for σ(n) as we defined in class. 2. The attempt at a solution For Part 2, I...

Homework Statement Prove that the product of any three consecutive natural numbers is divisible by 6. Homework Equations The Attempt at a Solution wat

I believe that Fibonacci primes are infinite. Currently there is no proof that there is an infinite number of Fibonacci primes. I was wondering why we couldn't compare the set of Fibonacci primes to the set of Natural Numbers and demonstrate that both have cardinality aleph null? Indeed, why...

Homework Statement For every epsilon > 0, there exists an N\in N such that, for every j >= N, |f(i,n) - g(n)|<epsilon for every n\in N. In addition, for every fixed j\in N, (f(i,n)) converges. Prove that (g(n)) converges. Homework Equations f: N x N --> R, g: N --> R The Attempt at...

\ Is \ \mathbb{N} \ dense \ in \ itself.

How can i find a bijection from N( natural numbers) to Q[X] ( polynomials with coefficient in rational numbers ). I can't find a solution for this. Can you please point me in the right direction ?

How can I solve that type of equation: x^2+y^2=4z^2 or x^2+3y^2=4z^2

How can i solve that equation: x^2 + y^2 = z^2-1 or x^2 + 3y^2 = z^2?

Since my late interests have been related to networks, I've started a pet project focusing on natural numbers network. I wanted to share my early explorations with this community since you have the proper background to interpret the results. I chose this network exactly because compared to...

Is Card (N even)< Card (N)? Where N even is set of all even Natural numbers, N is set of all Natural numbers. Hint: use the mapping from N eve→Nn N even to N is given by n-->n a. Show examples of this mapping from N even N even to N. b. Is the mapping above onto? One-to-one? My try...

Homework Statement Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty. (a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...

Homework Statement Show that if n belongs to N, and: An: = (1 + 1/n)^n then An < An+1 for all natural n. (Hint, look at the ratios An+1/An, and use Bernoulli's inequality) The Attempt at a Solution I think i have a vague idea of what to do here, like I am sure induction is involved in this...

Homework Statement Every triple of consecutive odd natural numbers, with the first being at least 5, contains at least on composite. Homework Equations N/A The Attempt at a Solution I know from number theory that of every set of consecutive odd integers, one of them is divisible...

Homework Statement The difference between cubes of two natural numbers is 208. Which are those two numbers?Homework Equations -The Attempt at a Solution Here is how I set it up. x\widehat{}^{}3 - (x-y)\hat{}3 = 208, which leads to ... x\hat{}2(3y) + x(-3y\hat{}2) + (y\hat{}3 - 208) = 0 I...

Homework Statement This is actually only related to a problem given to me but I still would like to know the answer. From my understanding, Cantor's Diagonalization works on the set of real numbers, (0,1), because each number in the set can be represented as a decimal expansion with an...

what is the convention you adhere to when it comes to natural numbers? for example there is a long standing debate about 0... should we define \mathbb N = \{0,1,2,...\} or instead \mathbb N = \{1,2,3,...\} and more about this, considering Peano's Axioms than we could choose \mathbb N...