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Questions tagged [natural-numbers]

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

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formal definition of operations in the ring of integers $\mathbb{Z}$

How do I compute step by step this formal definition of the multiplication operation $\cdot_{\mathbb Z}$ in $\mathbb Z$ $$(m,n)\cdot_{\mathbb Z}(p,q)=(m\cdot p+n\cdot q,m\cdot q+n\cdot p) ?$$ What ...
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Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite.

Let $n \in \omega$. Suppose $f:n \to A$ is onto $A$. Prove that $A$ is finite. I have: Let $I_a = \{i \in n:f(i)=a\}$ for $a \in A$. Since $f$ is onto $A$, $I_a$ is nonempty, and by the well-...
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Does the Axiom schema of Replacement imply the Axiom of Infinity? [duplicate]

The axiom of infinity says that the set of natural numbers exists, while the axiom of replacement says that if an object (a member of a set) exists, then all definable mappings of that object yield ...
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Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto.

Question: Prove that $p:\omega \times \omega \to \omega$ defined by $p(i,j)=2^i(2j+1)-1$ is one-to-one and onto. Then show that if $i\in m$ and $j \in n$, then $p(i,j) \in p(m,n)$. The book offers a ...
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For any natural $m \neq n$ show that $|\sqrt[n]{m} - \sqrt[m]{n}| > \frac{1}{mn}$.

Here's my try. The inequality above is equivalent to $$|m^{\frac{1}{n}} - n^{\frac{1}{m}}|> \frac{1}{mn}$$ First, I want to get rid of the absolute value. Assume without loss of generality that $...
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$f,g: \mathbb{N} \rightarrow \mathbb{N} $, where $f(n)=g(2n)$. Prove that if $f$ is surjective, $g$ is not injective. [duplicate]

Trying this question for a while. $f,g: \mathbb{N} \rightarrow \mathbb{N} $ $\forall n \in \mathbb{N}: f(n)=g(2n)$ I need to prove that if $f$ is surjective $g$ is not injective. Now I see that $g$...
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Q: Prove that for any natural number, there exists a multiple that (in decimal form) only uses digits 0 and 1 [duplicate]

I'm supposed to prove the theorem in the title for a combinatorics class (continuation of discrete structures class). At the moment I have no idea how to aproach this question. I would appreciate ...
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$3n = 8x+4$ control flag inside of a function

I want to be able to evaluate wether the result of a function satisfies an expression. Is this possible to do? Example I have an expression, a linear polynomial: $$8x+4$$ for $x\in\mathbb{N}$ If I ...
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Proof that square of sum equal to sum of cubes of N. [duplicate]

Does exist a proof for this: $$(1 + 2 + 3 + 4)^2=1^3 + 2^3 + 3^3 + 4^3$$
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1answer
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Prove that for every $n$, the binary representation of $n + 1$ contains exactly one bit that flips from $0$ to $1$

I know since $n$ is a binary representation it can be represented $ \sum_{i = 0}^{p}b_{i}\cdot 2^{i}$, where $b_{i}\in \{0,1\}.$ I have the intuition for this problem, i think. If $n$ is odd then the ...
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Maximum standard deviation of $n$ positive integers with set mean

Suppose the set $S$ contains $n$ positive integers. If the mean $\mu$ of the elements is known, is there a method to finding the maximum possible value of the standard deviation $\sigma$ for $S$? I ...
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Positive integers that are equal to the product of their figures [closed]

How many are the natural numbers (whole positive integers ≥ 1) that are equal to the product of their figures?
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Why is absolute value of negative exponent equal to positive value?

I am asked to integrate the following: $\int_{-\infty}^{0}e^{-\left\lvert 3x\right\rvert}dx$ And I am told that $e^{-\left\lvert x\right\rvert}=e^{x}$ How is it that an absolute value (the exponent)...
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3answers
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Why is the derivative of $3^x$ equal to $3^x \cdot \ln 3$

Our teacher tells us to convert it this way $ 3^x = e^{\ln 3^x}= e^{x\cdot\ln 3}$ and then use the rule $e^u\cdot u'$ but I can't understand where $\ln$ comes from and how $\ln 3^x$ = $x\cdot \ln 3$.
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Let $F:A \to A$ be a function and let $Y \subseteq A$. Prove that the class $S = \{B:Y\subseteq B \subseteq A$ and $F[B] \subseteq B\}$ is a set.

Let $F:A \to A$ be a function and let $Y \subseteq A$. (a) Prove that the class $S = \{B:Y\subseteq B \subseteq A$ and $F[B] \subseteq B\}$ is a set. Can I use the theorem that states: Let $\phi(x)$ ...
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2answers
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Prove that $n\not=n^+$ for all $n \in \omega$

Prove that $n\not=n^+$ for all $n \in \omega$. Where $\omega$ is the set of natural numbers, and $n^+ = n ∪ \{n\}$. Attempt: Assume $n = n^+$ for some $n \in \omega$. Then $n=n∪\{n\}$. This is ...
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Unique form to write any natural number

I want to know if this is true, I have done many examples, but don't know how to prove it: If $m \geq 1$ and $0 \leq k <m$, then, for any $n \in \mathbb{N}$, there is a unique $m$ and $k$ such ...
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If $y$ is an integer and $y^n= k^{\gamma}\beta$, $\gcd(\beta,k)=1$, then $k^{\gamma/n}$ and $\beta^{1/n}$ are integers [duplicate]

Proof: Since $\gcd(\beta, k)=1$, we get $\gcd(\beta, k^\gamma)=1$. Now, $y$ is an integer. So, $(y^n)^{1/n}=k^{\gamma/n}{\beta}^{1/n}$ must be an integer. But, there is no common prime between $k^{\...
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Mapping $\Bbb N\to\Bbb Q$

I want to set up a map from $\Bbb N\to\Bbb Q.$ Take $\Phi_S(x)=e^{(S/\ln(1-x))}$ and $M_T(1-x)=\Phi_S(x); S,T\in\Bbb N.$ Set $\Phi_S(x)=M_T(x)$ to obtain algebraic $x$ coordinates. If $x$ happens ...
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What does $(a_n)_n \in A^{\mathbb{N}}$ mean?

What does $(a_n)_n \in A^{\mathbb{N}}$ mean? What kind of sequence is that? How does the indexing work? What's the $A$ to natural numbers power?
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Official name of “sized integers”, a kind of number where $00$ is not equal to $0$?

Hierarchical identifiers, labels and indexes... All can use digits as character-strings, differenciating $0$ and $00$, $1$ and $001$, but preserving all other numeric interpretations, like order ($...
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Is there any way to prove that $x = y \Rightarrow x + z = y + z$?

Terence Tao, Analysis I, 3e, A.7 Equality (...) How equality is defined depends on the class T of objects under consideration, and to some extent is just a matter of definition. However, ...
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Is possible to represent natural number X without 'splitting' it as additive-finitely repeated element set property?

I have a $Set$ of elements $n$ (a multiset). Each element of this set $n$ is the same, is a multiset because one element $n$ is replicable using addition operation, in other words we can have finitely ...
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1answer
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Is Halmos' use of the Axiom of Substitution wrong in Ch. 19 Naive Set Theory?

I am trying to work out how the axiom of substitution on the set $\omega $ of all natural numbers can be used to construct the set of all successors of $\omega$ (as in the set {$\omega,\omega^+,(\...
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Complex number vs Complex exponential

I know that a complex number is a point in 2D plane. I wonder how to describe what is a complex exponential?
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Why doesn't the construction of $\mathbb{N}$ through ordinals in ZFC violate Gödel's Incompleteness Theorem?

The title kind of says it all. I've been working through Axiomatic Set Theory, Suppes and Mathematical Logic, Kleene. And I haven't thoroughly studied ordinals and incompleteness yet. But, ...
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Show $(N, ≤)$ is a partial order with least upper bounds (lubs) and greatest lower bounds (glbs) of all pairs.

does anyone have any solution or a good hint? Let $(\mathbb{N}, ≤)$ be the set of natural numbers with the relation $m ≤ n$, meaning $m$ divides $n$. Show $(\mathbb{N}, ≤)$ is a partial order with ...
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1answer
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Proof of Natural Numbers using n+1 = n ∪ {n}

In set theory natural numbers are defined by 0 = ∅ and natural number n+1 = n ∪ {n} I need to prove that for every n ∈ N , n = {k ∈ N | k < n}. I know that natural numbers 1 = {∅} 2 = {∅,{∅}} ...
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Need help finishing proving this summation inequality by induction

I'm having a hard time solving the following problem: Prove by induction that $\forall\ n\in\mathbb{N} $: $$\sum_{i=1}^{n} \frac{n+i}{i+1}\ge n$$ I checked the base case, and then made it up to ...
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1answer
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Natural Number Inductive Proof

Prove the following statement: For every $\lambda$>1, there exists a number a∈N and b∈[0,1) such that $\lambda$=a+b. I first defined a = sup{n ∈ N | n ≤ x}, so m is the integer part of x or the ...
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Closed natural numbers

I am reading this paper about interleavers for turbo code design, and when it describes the so called block interleavers, it says that To obtain a block interleaver function it is necessary to ...
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Is there a natural number satisfying the given condition? [duplicate]

Is there a positive integer $n$ such that $\sum_{k=0}^{n}\sqrt{n+k}$ is also an integer?
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Show that if $2^k\in S\space\space\forall k\in\mathbb{N}$, and if $k\in S$ and $k\ge2$, then $k-1\in S$, then $S=\mathbb{N}$

Show that if $2^k\in S\space\space\forall k\in\mathbb{N}$, and if $k\in S$ and $k\ge2$, then $k-1\in S$, then $S=\mathbb{N}$ My attempt: Let $n\in\mathbb{N},$ hence $2^n\in\mathbb{N}$ by the first ...
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Proving that $\mathbb{N}$ isn't bounded from above using Bolzano-Weierstrass

Here's my attempt but I don't think it's good enough: We suppose thar $\mathbb{N}$ is bounded from above (we already know it's bounded from below). Let $a_n$ be the sequence of natural numbers, $n\in ...
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1answer
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Using Induction to prove this inequality?

I'm having a hard time solving the following problem: Let ${\displaystyle (a_{n})_{n\in \mathbb {N_{0}} }}$ be the sequence defined recursively by: $a_{0}=2$ $a_{n+1} = \frac{1+3^{n+1}}{2} + \sum_{...
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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" ...
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2answers
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Using Induction to prove a product series equality? [duplicate]

I'm getting confused by these types of problems where "n" appears in the general term: Use Induction to prove: $$\prod_{i=1}^n \frac{n+i}{2i-3}= 2^n(1-2n)$$ Would my P(n+1) be this?: $\mathbf P(n+...
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1answer
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How to show $n=1+\sum_{k=1}^{n}\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor$ for every natural number $n$.

While answering a question here I noticed that: $$n=1+\sum_{k=1}^{n}{\left\lfloor{\log_2\frac{2n-1}{2k-1}}\right\rfloor}$$ for every natural number $n$. I tried to demonstrate it using Legendre's ...
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0answers
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Writing down consecutive natural numbers until a certain number of digits $k$ is reached.

A person starts writing consecutive natural numbers from $5$ until $k$ digits are reached. For some values of $k$, this will be impossible, for example $6$ or $8$ are impossible as then after writing ...
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1answer
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Determine all $a$ and $b$ natural numbers such that $\frac {a^2+2b} {b^2-2a}$ and $\frac {b^2+2a} {a^2-2b}$ are whole numbers.

I proceeded in the following way: It is clear that $a \ne 0$ and $b \ne 0$. Let $\frac {a^2+2b} {b^2-2a} = k, k \in \mathbb{Z} \tag 1$ and $\frac {b^2+2a} {a^2-2b} = m, m \in \mathbb{Z} \tag 2$ ...
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Do two pairs of distinct natural numbers exist such that AGM(A,B) equal to AGM(C,D)?

Here AGM is arithmetic-geometric mean. Are there natural numbers A,B,C,D such that $1\leq A<C<D<B$ and arithmetic-geometric mean AGM(A,B)=AGM(C,D) ? In other words, is AGM a homomorphism ...
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Can arithmetic-geometric mean of two distinct natural numbers be a natural number?

Are there solutions to equation $AGM(A,B)=C$ such that $1\leq A<B$ and $A,B,C\in\mathbb{N}$?
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How should we parse this proof that every infinite set has a subset which is eqivalent to the set of natural numbers?

This is again from the chapter Construction of the System of Real Numbers in The Fundamentals of Mathematics, Volume 1. It may be the case that the original wording in the German language would be ...
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1answer
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Is this sufficient to prove the associativity of natural number addition?

This question is regarding the same proposition as is discussed in How should this proof of the associativity of natural number addition be understood? Here I wish to ask if a much simpler proof than ...
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29 views

Find the natural number which is divisibled by another number

Let $n= \overline{a_1a_2a_3}$ ($a_1 \neq a_2$, $a_2 \neq a_3$, $a_3 \neq a_1$). 1/ How many possible values does $n$ have which is divisibled by $7$? 2/ How many possible values does $n$ have which ...
3
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1answer
107 views

What class of natural numbers known to be infinitely large occurs least frequently? [closed]

By "class" I mean sets of natural numbers with a given specific property (i.e. prime numbers or perfect numbers). Obviously all infinitely large sets have the same "size", but for example solitary (as ...
0
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1answer
65 views

How many $10-$digit numbers are divided by $11.111$ and all the digits are different?

The Problem: How many $10-$digit numbers are divided by $11.111$ and all the digits are different? A) $3250$ B) $3456$ C) $3624$ D) $3842$ E) $4020$ The Problematic point ...
4
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2answers
101 views

If a prime $p$ divides $n^2$ then it also divides $n$ - Is this proof correct?

I'm learning Real Analysis by myself and I wanted to prove that if a prime $p$ divides $n^2$ where $n$ is an integer, then $p$ divides $n$ itself. I saw that proving this is the same as saying that ...
6
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1answer
74 views

Define a model for $\mathbb N$ without set theory

I've been looking around for a long time about how to found mathematics on a solid base. This led me to a long and painful journey of avoiding circular loops. It led me to do a bit of elementary ...
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1answer
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$\omega^\omega$ 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 ...