Questions tagged [natural-numbers]

For question about natural numbers $\Bbb N$, their properties and applications

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of and , respectively (see English numerals). A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to , such as the distribution of , are studied in . Problems concerning counting and ordering, such as partition enumeration, are studied in .

1107 questions
-1
votes
2 answers

When extending the natural numbers to the integers when is it legal to set a natural number equal to an integer.

My source BBFSK I need to add that natural numbers in this context are defined as starting with 1. I didn't think that would impact the answer, but apparently it does. $n-0$ provides a "bridge" between the integers and the natural numbers. This…
-1
votes
1 answer

$\omega^\omega$ correspondence with $\mathbb R$-irrationality

Here in the second comment I do not understand why $\omega^\omega$ corresponds to irrational numbers? : In my experience one typically identifies $ω^ω$ with the irrational elements of R; and then we call them "reals" because they are equinumerous,…
user122424
  • 3,598
  • 2
  • 15
  • 26
-1
votes
1 answer

Complicated index models and Boolean algebras/ Shelah/ Unclear step in the proof

Here on the page $10$ in the $5$th line (the proof of lemma $1.10$), Shelah defines $n_*$ as $\omega$: $$n_*=\omega,$$ and then he continues: be such that $n_*\geq\text{max}\{n(0),...,n(m-1)\}<\omega.$ Does it make sense?
-1
votes
1 answer

$f(x,y)=(x/2^y) \mod 16$ a Bivariate function?

I have a two-input function on the integers or naturals. Is a bivariate function a function that takes two inputs or is there anything more to it? For example take the function: $f(x,y)=(x/2^y) \mod 16$ where $f:\mathbb{N}\to\mathbb{N}$ or…
Natural Number Guy
  • 1,084
  • 1
  • 13
  • 27
-1
votes
1 answer

Find the natural numbers such that a number is a prime number.

Find the natural numbers $x$, $n$ such that $p = x^4+2^{4n+2}$ is prime number.
-1
votes
1 answer

Divisor Function of Sums in Fractions

I have a question that I've been working on for a while now. It says, "Let $A=\{0,1,2,\dots,2018\}$. Prove that $\forall n\in\mathbb{N},\exists\{a_0,a_1,a_2,\dots,a_{2018}\}\subseteq\mathbb{N}$,…
-1
votes
2 answers

How to show that $(S\cup\{0\},\ge)$ is order-isomorphic to $(S,\ge)$?

Let $S$ denote the set containing all the natural numbers that are not divisible by $ 2 $. And define the binary relation $ \ge $ on two natural number $ m , n $ , $m \ge n $ if $m = k n $ , for integer $k$, meaning that $ n $ divides $ m $. How to…
user536852
-1
votes
1 answer

Suppose m, n ∈ N. Explain why n|m implies n ≤ m.

I know that, by definition, n|m implies m=pn for which p is an integer, but I don't know how to get n ≤ m.
-2
votes
1 answer

$-\mathbb{N}$ is not bounded below, right?

Parable: I am asked to prove by contradiction that the set of negative integers $\mathbb{N}$ is not bounded below. My professor writes $$-\inf(-\mathbb{N})=\sup(\mathbb{N}),$$ and says that since $LHS\in\mathbb{R}$ and $RHS\not\in\mathbb{R}$, by the…
Secure Space
  • 141
  • 2
  • 9
-2
votes
0 answers

Convex games and a convex function $f:\mathbb{N}\to\mathbb{R}$

I'm stocked with this exercise in Game Theory... A function $f:\mathbb{N}\to\mathbb{R}$ is called convex if $\forall i,j,k\in\mathbb{N}$ such that $i\le j\le k$ and $j
-2
votes
1 answer

How many natural numbers can't be written as 5p+7q, where p and q are natural numbers?

If $A = \{5p+7q|p,q \in \mathbb{N}\}$, determine $|\mathbb{N} \setminus A|$.
-2
votes
3 answers

I need help with the proof of Theorem 4, Chapter I in E. Landau's "Foundations of Analysis"

I need help with the proof of Theorem 4, Chapter I in E. Landau's "Foundations of Analysis" To every pair of natural numbers $x$, $y$, we may assign in exactly one way a natural number, called $x+y$ such that: $x+1=x'$ [where $x'$ has to be…
Matteo
  • 43
  • 2
-2
votes
1 answer

Supremum, infimum, max and min for $n(1+(-1)^{n})$

My question is, if my assumption is correct, for the set $$A = \{ n(1+(-1)^{n}) : n \in \mathbb{N},n\geq 1\}$$ I think that when $n$ is odd we have $\sup A = 0, \inf A= 0, \max A = 0, \min A = 0 $, and for even $n$ we have $ \sup A= +\infty, \inf A=…
-2
votes
1 answer

$Q(n)-Q(n-1) = T(n)$ Prove that $Q(n)$ degree is $k+1$

I was given this problem and I've been thinking a lot of time and still I have nothing. $Q:ℕ↦ℕ$ $Q(n)-Q(n-1) = T(n)$ $T(n)$ degree is $k$ Prove that $Q(n)$ degree is $k+1$ any idea? Thank you
-2
votes
1 answer

"Proof" of $0=1$ in set theory

Ok, so here is a proof of "$0 = 1$" I came up with today. You can do in set-theory, where natural numbers are defined in the usual way. Proof: Let $\mathsf{Succ}$ be the function that takes any natural number and adds one to it. Then we have…
Léreau
  • 2,748
  • 1
  • 11
  • 32
1 2 3
73
74