Show that every two blocks have at most one vertex in common

granuliet1u
2022-08-03
Answered

Show that every two blocks have at most one vertex in common

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asked 2022-06-29

Recall that Bertrand's postulate states that for $n\ge 2$ there always exists a prime between $n$ and $2n$. Bertrand's postulate was proved by Chebyshev. Recall also that the harmonic series

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $

and the sum of the reciprocals of the primes

$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots $

are divergent, while the sum

$\sum _{n=0}^{\mathrm{\infty}}\frac{1}{{n}^{p}}$

is convergent for all $p>1$. This would lead one to conjecture something like:

For all $\u03f5>0$, there exists an $N$ such that if $n>N$, then there exists a prime between $n$ and $(1+\u03f5)n$.

Question: Is this conjecture true? If it is true, is there an expression for $N$ as a function of $\u03f5$?

$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots $

and the sum of the reciprocals of the primes

$\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots $

are divergent, while the sum

$\sum _{n=0}^{\mathrm{\infty}}\frac{1}{{n}^{p}}$

is convergent for all $p>1$. This would lead one to conjecture something like:

For all $\u03f5>0$, there exists an $N$ such that if $n>N$, then there exists a prime between $n$ and $(1+\u03f5)n$.

Question: Is this conjecture true? If it is true, is there an expression for $N$ as a function of $\u03f5$?

asked 2022-06-15

I have been recently studying a C.G. Hempel's article on mathematical truth and pointed out his following quotation: "Every concept of mathematics can be defined by means of Peano's three primitives,and every proposition of mathematics can be deduced from the five postulates enriched by the definitions of the non-primitive terms".

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)

I was wondering if it is possible for someone to make a scheme illustrating the sequence of the derivation of the whole theory of mathematics being derived by these postulates.Possibly starting from natural numbers?(notice that Hempel excludes geometry)

asked 2022-05-09

Let $A$,$B$ be two points on opposite sides of a line $l$. Then the line segment $AB$ intersects $l$.

My question is:

1. Using only the 4 postulates of Euclid, is there a way to make precise the meaning of ``opposite sides"?

2. Is the intersection guaranteed to exist using only the 4 postulates? If so, why is it true?

My question is:

1. Using only the 4 postulates of Euclid, is there a way to make precise the meaning of ``opposite sides"?

2. Is the intersection guaranteed to exist using only the 4 postulates? If so, why is it true?

asked 2022-08-31

The necessary and sufficient condition for a non - empty subset W of a vector space V(F) to be a subspace of V is

a,b in F and $\alpha $, $\beta $ in W implies a$\alpha $ + b$\beta $ in W

I need to prove the postulates of vector space with this condition. Hints ?

a,b in F and $\alpha $, $\beta $ in W implies a$\alpha $ + b$\beta $ in W

I need to prove the postulates of vector space with this condition. Hints ?

asked 2022-05-29

I want to show that $2{p}_{n-2}\ge {p}_{n}-1$...

Bertand's postulate shows us that $4{p}_{n-2}\ge {p}_{n}$ but can we improve on this?

any ideas?

Bertand's postulate shows us that $4{p}_{n-2}\ge {p}_{n}$ but can we improve on this?

any ideas?

asked 2022-06-26

By Bertrand's postulate, we know that there exists at least one prime number between $n$ and $2n$ for any $n>1$. In other words, we have

$\pi (2n)-\pi (n)\ge 1,$

for any $n>1$. The assertion we would like to prove is that the number of primes between $n$ and $2n$ tends to $\mathrm{\infty}$, if $n\to \mathrm{\infty}$, that is,

$\underset{n\to \mathrm{\infty}}{lim}\pi (2n)-\pi (n)=\mathrm{\infty}.$

Do you see an elegant proof?

$\pi (2n)-\pi (n)\ge 1,$

for any $n>1$. The assertion we would like to prove is that the number of primes between $n$ and $2n$ tends to $\mathrm{\infty}$, if $n\to \mathrm{\infty}$, that is,

$\underset{n\to \mathrm{\infty}}{lim}\pi (2n)-\pi (n)=\mathrm{\infty}.$

Do you see an elegant proof?

asked 2022-05-09

I came up with this question- how would you show 4 not equal to 6 (or m not equal to m+n ( n not 0)), using only Peano's Postulates?

I can see a number of things go wrong- for instance the Principle of Mathematical Induction seems to fail. Also possibly 0 seems to be in the image of the successor function in that case.

I can see a number of things go wrong- for instance the Principle of Mathematical Induction seems to fail. Also possibly 0 seems to be in the image of the successor function in that case.