instruction stringlengths 12 30k |
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Could someone tell me what the "square of a graph" $G^2$ is? Thanks. |
**HINT**: Find out what points $(x, y)$ with $x = 1$ also have distance $2$ to the origin. What would these points represent? |
I'm not going to prove something, this is just a question. From the first day which I went to University until now, I'm always thinking that the whole math science is based on false assumption( I'm sure it's not true, please help me know why ? ). Why ?
**For example:** ***infinity*** !
What is infinity ? Is that ... |
Does really infinity and zero exist ? |
How can I calculate average distance from square center to points inside the square? |
What is average distance from square center to some point? |
Do infinity and zero really exist ? |
is integral of (1/x dx) the same as (dx/x) ?? |
Suppose we have the following two identities:
- $\displaystyle \sum_{k=0}^{n} \binom{n}{k} = 2^n$
- $\displaystyle \sum_{k=0}^{n} (-1)^{k} \binom{n}{k} = 0$
The first says that the number of subsets of an $n$-set is $2^n$. The second says that the number of subsets of even size equals the number of subsets of... |
The second identity can be proven by an inclusion-exclusion argument, but maybe it's clearer this way: pick an arbitrary element of the $n$-element set. Given a subset, if it contains the arbitrary element, remove it; otherwise, add it in. (That is, take the symmetric difference.) This defines a bijection from the s... |
Do you know a [Pairing function][1]? Do you see how this can help?
[1]: http://en.wikipedia.org/wiki/Pairing_function |
It seems like your premise for mathematics being a 'lie' is that its theoretical background cannot be completely applied to the real world (e.g. having the smallest distance in the real world, but not in mathematics). If that's how you define a lie, sure, but mathematics is an abstract science and I am not sure why you... |
Say I have a time series, with actual measurements of a variable $a$ in different locations. If I want to know the average of $p$ from time $t_1$ to $t_n$, I could say that $\overline{a_{p}}=\frac{\sum\limits_{t=1}^n a_{{p}_t}}{n}$, where $p$ is the coordinate of the point of measurement. So far, so good. Let's gene... |
[This article](http://en.wikipedia.org/wiki/Infinity) on Wikipedia states the following:
>Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indee... |
Is $[0;1]$ equinumerous to $\lim_{x \to \infty} $ $R^x$? |
I think your questions and struggles with these difficult concepts are completely understandable, but your presentation is terrible. You are unnecessarily confrontational and aggressive. Having difficulties with these concepts is completely all right. Assuming that during hundreds of years mathematicians conspired to b... |
From wikipedia,
http://en.wikipedia.org/wiki/Riemann_zeta_function
"Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis."
I know the zeros are symmetrical. But what ... |
[This article](http://en.wikipedia.org/wiki/Infinity) on Wikipedia states the following:
>Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indee... |
Is $[0;1]$ equinumerous to $\mathbb{R}^{\omega}$? |
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists
an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is: Using this defini... |
Why is the mobius strip not orientable? |
What is $\lim\limits{x\to 0}\frac{\lceil x\rceil}{x}$ ? Here, $\lceil x \rceil$ is the ceiling function at $x$.
For left limit and right limit as $x\to 0$.
Thanks in Advance . |
what is $\lim\limits_{x\to 0} [x]/x$ ? |
what is $\lim\limits_{x\to 0} \lceil x\rceil/x$ ? |
I am trying to understand the notion of an orientable manifold.
Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1}))> 0$ (where defined). My question is:
Using this def... |
Why is the Möbius strip not orientable? |
Let M be a smooth manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ such that for all $\alpha, \beta$, $\textrm{det}(J(\phi_{\alpha} \circ \phi_{\beta}^{-1})) > 0$ (where defined). I'm struggling to understand the reason this definition is made. My question... |
How should I think about what it means for a manifold to be orientable? |
Get two intersecting charts $(U,\phi)=(U, (x^i))$, $(V,\psi)=(V, (y^i))$, with $p\in U\cap V$. Then there are two basis for the tangent space of $M$ at $p$: $\{ \frac{\partial}{\partial x^\alpha}\}$ and $\{\frac{\partial}{\partial y^\beta}\}$ . Intuitively, these two bases have the same orientation (in the vector space... |
What is the value of $\lceil x\rceil$ for small positive $x$? Think of $0<x<1$. |
$\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$
[This is what the Alpha Wolf says][1].
[1]: http://www.wolframalpha.com/input/?i=lim+(Ceiling[x]/x)+as+x-%3E0 |
$\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$
[This is what the Alpha Wolf says][1]:
> $\lim \limits_{x \to 0^-} \frac{\lceil x \rceil}{x} = 0$
>
> $\lim \limits_{x \to 0^+} \frac{\lceil x \rceil}{x} = \infty$
The two limits aren't equal. And what could this possible mean?
[1]: http://www.wolframa... |
$\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$
[This is what the Alpha Wolf says][1]:
> $\lim \limits_{x \to 0^-} \frac{\lceil x \rceil}{x} = 0$
>
> $\lim \limits_{x \to 0^+} \frac{\lceil x \rceil}{x} = \infty$
The two limits aren't equal. And what could this possible mean?
How did we get this? Look ... |
$\lim \limits_{x \to 0} \frac{\lceil x \rceil}{x}$
[This is what the Alpha Wolf says][1]:
> $\lim \limits_{x \to 0^-} \frac{\lceil x \rceil}{x} = 0$
>
> $\lim \limits_{x \to 0^+} \frac{\lceil x \rceil}{x} = \infty$
The two limits aren't equal. And what could this possible mean?
How did we get this? Look ... |
I'm reading a book about Calculus on my own and am stuck at a problem, the problem is
> There are two circles of radius 2 that have centers on the line x = 1 and pass through the origin. Find their equations.
The equation for circle is $(x-h)^2 + (y-k)^2 = r^2$
Any hints will be really appreciated.
**EDIT:*... |
Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the lenght of an a-string through b is at most 4.
What is the biggest scenery X such that:
1) $a+b$ is a root and $a+2b$ is not a root?
2) $a+2b$ is a root and $a+3b$... |
Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the lenght of an $a$-string through $b$ is at most 4.
What is the biggest scenery X such that:
1) $a+b$ is a root and $a+2b$ is not a root?
2) $a+2b$ is a root and $a... |
Suppose the universe has shape $M$. Then $M$ is orientable if and only if you can write yourself a note, then begin a journey, and come back to always find that you can still read the note. If $M$ is not orientable, it is possible to return from a journey to find that the note is written backwards and, indeed, that e... |
Generalized tractrices (tractories) exist, see [this][1] or [this][2] or [this][3]. (the last two are in French, but with slightly more detail than the first one.)
This [old book][4] ought to be of interest as well.
This [book][5] describes Euler's treatment of the problem of the tractory.
As a note, the probl... |
Let $a,b$ be roots ($a\ne \pm b$) of a Lie algebra of type $X$, where $X$ can be classic or exceptional $(A,B,C,D,E,F,G)$. It is well known that the length of an $a$-string through $b$ is at most 4.
What is the biggest scenery X such that:
1) $a+b$ is a root and $a+2b$ is not a root?
2) $a+2b$ is a root and $a... |
First of all please agree my apologies because of my attitude and the way I asked my question. But believe me or not, whenever I asked this kind of questions from my teachers they down voted me in real world ;) . I should ask my question in more warily way. As Yuan commented in my question "I'm not going to get any upv... |
1. In [Wikipedia][1], well-order is
defined as a strict total order on a set
with the property that every
non-empty subset of the set has a least
element in this ordering.
But then [later][2], well-order is
defined as a total order on S with
the property that every non-empty
... |
1. In [Wikipedia][1], well-order is
defined as a strict total order on a set S
with the property that every
non-empty subset of S has a least
element in this ordering.
But then [later][2], well-order is
defined as a total order on S with
the property that every non-empty
... |
I have a polygon in 2D ( defined by a series of Vertex $V$ with coordinates). The polygon can be convex or concave. I have $n$ number of fix points I can put inside the polygon.
The question is, how can I distribute the fix points as uniformly as possible inside the polygon?
The motivation for this question is I... |
First of all please agree my apologies because of my attitude and the way I asked my question. But believe me or not, whenever I asked this kind of questions from my teachers they down voted me in real world ;) . I should ask my question in more warily way. As Yuan commented in my question "I'm not going to get any upv... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now, I'm always thinking that the whole math science is based on false assumption( I'm sure it's not true, please help me know why ? ). Why ?
**For example:** ***infinity*** !
What is infinity ? Is that ... |
First of all please agree my apologies because of my attitude and the way I asked my question. I changed it again. I really didn't aggressive in reality just present my question in a bad way. But believe me or not, whenever I asked this kind of questions from my teachers they down voted me in real world. I should ask m... |
I have a polygon in 2D ( defined by a series of Vertex $V$ with coordinates). The polygon can be convex or concave. I have $n$ number of fix points I can put inside the polygon.
The question is, how can I distribute the fix points as uniformly as possible inside the polygon?
The motivation for this question is I... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please help me to solve the most important of those problems.
**For example:** ***infinity*** !
What is infinity ? Is tha... |
The simplest idea is to generate a sequence of points $(X_i,Y_i)$, where $X_i,Y_i$ are independent uniform random variables in a rectangle containing the polygon. Then, you simply drop the points which fall outside your polygon. You continue until you get $n$ unrejected points.
As we could expect, this problem has b... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please help me to solve the most important of those problems.
**For example:** ***infinity*** !
What is infinity ? Is tha... |
The simplest idea is to generate a sequence of points $(X_i,Y_i)$, where $X_i,Y_i$ are independent uniform random variables in a rectangle containing the polygon (the minimal rectangle bounding the polygon). Then, you simply drop the points which fall outside the polygon. You continue until you get $n$ unrejected point... |
How do I correctly represent the following pseudocode in math notation?
if m = 1 then
if (a,b)_1 < (a,b)_n then
r = (a,b)_1 / (a,b)_n
else
r = (a,b)_n / (a,b)_1
else if m = 2 then
if (a,b)_1 < (a,b)_n then
r = (a,b)_1 / (a,b)_n
else
r = (a,b)_n... |
The simplest idea is to generate a sequence of points $(X_i,Y_i)$, where $X_i,Y_i$ are independent uniform random variables, in a rectangle containing the polygon (the minimal rectangle bounding the polygon). Then, you simply drop the points which fall outside the polygon. You continue until you get $n$ unrejected poin... |
How do I correctly represent the following pseudocode in math notation?
EDIT: Formula expanded.
if m = 1 then
if (a,b)_1 = (a,b)_n then
r = 1
else if (a,b)_1 < (a,b)_n then
r = (a,b)_1 / (a,b)_n
else
r = (a,b)_n / (a,b)_1
else if m = 2 then
if (a,b)_... |
How do I correctly represent the following pseudocode in math notation?
EDIT1: Formula expanded.
EDIT2: Clarification.
(a,b) represents a line segment on a 1D line. a <= b for each segment. The division show below is done as per the following T-SQL code (which I suppose could be represented as a function in the ... |
How do I correctly represent the following pseudocode in math notation?
EDIT1: Formula expanded.
EDIT2: Clarification.
(a,b) represents a line segment on a 1D line. a <= b for each segment. The division show below is done as per the following T-SQL code (which I suppose could be represented as a function in the ... |
the problem is the following:
For all x \in \mathbb{R}, we assign a finite set \Phi(x) \subset \matbb{R} - \{ x \}. We say that a set S \subset \mathbb{R} is independent if for any x,y \in S, x \notin \Phi(y) and y \notin Phi(x), that is, if S \cap \Phi(S) = \emptyset. Prove that there is an independent set S with t... |
the problem is the following:
For all $x \in \mathbb{R}$, we assign a finite set $\Phi(x) \subset \mathbb{R} - \{ x \}$. We say that a set $S \subset \mathbb{R}$ is independent if for any $x,y \in S$, $x \notin \Phi(y)$ and $y \notin \Phi(x)$, that is, if $S \cap \Phi(S) = \emptyset$. Prove that there is an independ... |
Given implicit function F(x,y) = 0 (it's hyperbola), how can I find its asymptotes? |
I am going to go out on a limb and recommend a more elementary book than (I think) any of the ones others have mentioned.
I claim that as a pure mathematician who is not a set theorist, all the set theory I have ever needed to know I learned from Irving Kaplansky's *Set Theory and Metric Spaces*. (And, you know, ... |
Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence
which is equidistributed in $[0,1)$.
(a) Is it true that $$\frac{1}{N}\sum_{n=1}^N f(x+\xi_n)$$ converges to the constant $\int_0^1 f(y) dy$
for each $x\in \mathbb{R}$ ?
(b) If so, is the convergence uniform over all $x$? |
Let $f$ be a function of period 1, Riemann integrable on [0,1]. Let $\xi_n$ be a sequence
which is equidistributed in $[0,1)$.
(a) Is it true that $$\frac{1}{N}\sum_{n=1}^N f(x+\xi_n)$$ converges to the constant $\int_0^1 f(y) dy$
for each $x\in \mathbb{R}$ as $N\to \infty$ ?
(b) If so, is the convergence unif... |
While I personally don't remember the mathematics, I can give you two leads which, if you follow it up, should give you some ideas.
- Try contacting [Jordan Ellenberg](http://www.math.wisc.edu/~ellenber/). Several years ago he gave a talk to the Princeton University Undergraduate Math Club on precisely what you de... |
Given implicit function F(x,y) = 0, how can I find its asymptotes?
EDIT: $F(X)=\sqrt{x^2 + y^2 + a} - \sqrt{x^2 + y^2 + b} - c$ |
Can you make a simple example of an uncountable ordinal? With <i>simple</i> I mean that it is easy to prove that the ordinal is uncountable. I know that the set of all the countable ordinals is an uncountable ordinal, but the only proof that I know is quite complicated. |
Given an abelian group $X$, let $F_n(X)$ denote the simplicial abelian group defined as follows:
$F_n(X)_j=0$ for all $j<n$ and $F_n(X)_j=X$ for all $j\geq n$
with the appropriate zero and identity maps between them so the normalization (normalized moore complex) gives the appropriate homology (say, $d_i=0$ for e... |
What's stopping me from choosing the nth Eilenberg Mac Lane space to be the following simplicial abelian group? |
Given implicit function F(x,y) = 0, how can I find its asymptotes?
EDIT: $F(x,y)=\sqrt{x^2 + y^2 + a} - \sqrt{x^2 + y^2 + b} - c$ |
Last month, I was introduced to the card game SET. The game raises several interesting questions eg what's the probability that n randomly drawn cards contain k sets, is it possible to end with 12 (or 15) cards which contain no sets... I would like to present the solution to a question like this to undergrads, and thro... |
If we only consider the natural numbers, is 3/2 undefined? If not what is the answer? |
Is 3/2 undefined when only considering the natural numbers? |
### Problem ###
Given:
$\vec r = r(sin \theta cos \phi, sin \theta sin \phi, cos \theta)$
$S: \left| \vec r \right| = a$, with $\hat n$ outwards
$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z)$
$\vec F = k \frac{\vec r - \vec r0}{ {\left| \vec r - \vec r_0 \right|}^3 }$
Calculate:
$\int_S \vec... |
Given implicit function F(x,y) = 0, how can I find its asymptotes?
EDIT: Sorry, my calculations were wrong. Here is correct function:
$F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} - c$ |
Show that $f_{n-1} + L_n = 2f_{n}$.
So we need to find a $2$ to $1$ correspondence.
Set 1: Tilings an $n$-board.
Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet.
So we need to decompose a tiling of an $n$-board to a tiling of an $n-1$-board or a tiling of an $n-1$-bracelet? |
Given implicit function F(x,y) = 0, how can I find its asymptotes?
EDIT: Sorry, my calculations were wrong. Here is correct function:
$F(x,y)=\sqrt{(x-a)^2 + (y-b)^2} - \sqrt{(x-c)^2 + (y-d)^2} - e$ |
Does f(n+1)/f(n) converge as n->infinity for f(n) defined by a linear recursion, for all linear recursions? |
Linear Recursion. Do ratio of consecutive terms converge for ALL linear recursions? |
Show that $f_{n-1} + L_n = 2f_{n}$.
So we need to find a $2$ to $1$ correspondence.
Set 1: Tilings an $n$-board.
Set 2: Tiling of an $n-1$-board or tiling of an $n$-bracelet.
So we need to decompose a tiling of an $n$-board to a tiling of an $n-1$-board or a tiling of an $n-1$-bracelet?
Source: *Proo... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please help me to solve the most important of those problems.
**For example:** ***infinity*** !
What is infinity ? Is tha... |
First of all please agree my apologies because of my attitude and the way I asked my question. I changed it again. I really didn't aggressive in reality just present my question in a bad way. But believe me or not, whenever I asked this kind of questions from my teachers they down voted me in real world. I should ask m... |
First, let me say why all axiomatic treatments insist on talking about homology of pairs. One of the (most important/useful?) axioms is the Long exact sequence axiom, for any pair $A$ contained in $X$ we get the long exact sequence of homology groups. This is crucial for making computations. The concept of the homology... |
1. The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula
$$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$
For small values of $n$, we have
$$V_1=2\qquad$$
$$V_2\approx 3.14$$
$$V_3\approx 4.18$$
$$V_4\approx 4.93$$
$$V_5\approx 5.26$$
$$V_6\approx 5.16$$
$$V_7\approx 4.72$$
It is not... |
Volumes of n-balls: what is so special about n=5? |
Given Kn define sequence S : writing all vertices in order of traversing it by Euler circle
how many distinct Hamilton cycles are in this sequence S? |
It can certainly vary. As an Eulerean path uses all n! edges and a Hamiltonian path uses n, you can't have more than (n-1)!. Can you have that many? There aren't any Eulerean paths if n is even and >2. Depending upon your definition, I can define an Eulearean path that has no segment that is Hamiltonian |
How about f(n+1)=f(n-1) started with f(0)=0, f(1)=1? |
Let $S=\{(x_1,x_2, \ldots x_n) \colon ~ f(x_1,x_2, \ldots x_n)=1,~ f:\mathbb {R^n} \to \mathbb R,~ f \text{ is } C^{\infty} \} \subseteq \mathbb {R^n}$.
>Is $S$ path-connected? If the answer for this is too advanced, what if $f$ is a polynomial over $\mathbb Q$.
**Context**: I had to establish that the set of dia... |
Let $S=\{(x_1,x_2, \ldots x_n) \colon ~ f(x_1,x_2, \ldots x_n)=1,~ f:\mathbb {R^n} \to \mathbb R,~ f \text{ is } C^{\infty} \} \subseteq \mathbb {R^n} \quad n \geq 2$.
>Is $S$ path-connected? If the answer for this is too advanced, what if $f$ is a polynomial over $\mathbb Q$.
**Context**: I had to establish that... |
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function with $n\geq 2$, and define $S_{f}=\{(x_1,x_2, \ldots x_n) \in\mathbb{R}^n \mid f(x_1,x_2, \ldots x_n)=1\} = f^{-1}(1)$.
> Is $S_f$ path-connected for any $f$ is $C^{\infty}$? If not, what about if $f \in \mathbb{Q}[x_1,\ldots,x_n]$?
**Context**: I had to establ... |
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function with $n\geq 2$, and define $S_{f}=\{(x_1,x_2, \ldots x_n) \in\mathbb{R}^n \mid f(x_1,x_2, \ldots x_n)=1\} = f^{-1}(1)$.
> Is $S_f$ path-connected when $f$ is $C^{\infty}$? If not, what about if $f \in \mathbb{Q}[x_1,\ldots,x_n]$?
**Context**: I had to establish... |
### Problem ###
Given:
$\vec r = r(sin \theta cos \phi, sin \theta sin \phi, cos \theta)$
$S: \left| \vec r \right| = a$, with $\hat n$ outwards
$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z) = \frac{3a}{5}(1,1,-1)$
$\vec F = k \frac{\vec r - \vec r0}{ {\left| \vec r - \vec r_0 \right|}^3 }$
Cal... |
Let $\sigma=(1\ 3),\ \tau=(2\ 4\ 5),\ \pi=(2\ 3\ 4) \in S_{5}$. Find $\pi\circ\tau\circ\sigma$.
I know the solution is $(1\ 4\ 5\ 3)$.
What i'm doing now is writing the permutations in the "expanded" form,
for example $\tau=\begin{pmatrix}
1\ 2\ 3\ 4\ 5\\
1\ 5\ 3\ 2\ 4
\end{pmatrix}$,
and then "f... |
Can someone explain the algorithm for composition of cycles? |
In the [wikipedia][1] article, two examples are given which use/ do not use the axiom of choice. They are:
- Given an infinite pair of socks, one needs AC to pick one sock out of each pair
- Given an infinite collection of pairs of shoes, one shoe can be specified without AC by choosing the left one.
Aren't ... |
In the [wikipedia][1] article, two examples are given which use/ do not use the axiom of choice. They are:
- Given an infinite pair of socks, one needs AC to pick one sock out of each pair.
- Given an infinite collection of pairs of shoes, one shoe can be specified without AC by choosing the left one.
Aren't... |
Yes, the "classification" of naturals by their **parity** (remainder modulo $2\:$) extends immediately to integers. Namely, *even* integers are those divisible by $2$, i.e. $\rm\: n = 2\:m\equiv 0\ (mod\ 2)\:,\: $ and *odd* integers are those leaving remainder $\:1\:$ when divided by $\:2\:,\ $ i.e. $\rm\ n = 2\:m + 1\... |
The idea behind the axiom of choice is to tell you how to choose when you can't necessarily distinguish between the items.
When you take out a pair of socks from the closet (or drawer, etc) you can't tell which one you had on your left foot and which on your right, or which one is this and which one is that. If you ... |
Yes, the "classification" of naturals by their **parity** (remainder modulo $2\:$) extends immediately to integers. Namely, *even* integers are those divisible by $2$, i.e. $\rm\: n = 2\:m\equiv 0\ (mod\ 2)\:,\: $ and *odd* integers are those leaving remainder $\:1\:$ when divided by $\:2\:,\ $ i.e. $\rm\ n = 2\:m + 1\... |
If I give you the following definition of the set $A$, how could you prove it is equal the set of the natural numbers without an explicit definiton for the latter?
The set $A$ is inductively defined as follows:
i) $0 \in A$; and <br/>
ii) $\forall n$, a natural number, if $n \in A$, then $n+1 \in A$.
I can ea... |
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