Let $\varphi:G \to S_{15}$ be a monomorphism. Let $P_2$ be a 2-subgroup. Its normalizer has $120/15 = 8$ elements, so the normalizer is just itself (wait whaaaat)
Let $\varphi:G \to S_{36}$ be the group action induced by the $36$ Sylow $5$-subgroups. Let $P$ be a Sylow $5$-subgroup. Then, $N(P)$ contains $5$ elements.
if $n_5=6$, then $\varphi:G \to A_6$ monomorphic, then $|N_G(H)| = 180/6 = 30$, then there is an element of order $3$, then there is an element of order $15$, and have fun embedding it into $A_6$
Hmm, date on the review is written in French, and the reviewer points out a sole reference that we should add, being a paper in French from 2008 (which is really late to still be publishing in French). My reviewer-recognition sense is tingling.
We have that $y-x>0$. By the Archimidean property we have that $n>\frac{1}{y-x}$. There also exists $a,b\in \mathbb{Z}$ such that $a<xn$ and $b>yn$. Why does the last sentence hold? From which property does this follow?
So, the Archimidean property? So, do we have that for $n\in \mathbb{N}$ and $\frac{a}{x}\in\mathbb{R}$ we get that $\frac{a}{x}<n\Rightarrow a<xn$ ? But how do we get then $b>yn$ ?
From the Archimedean Axiom we have that $\forall x\in \mathbb{R} \exists n\in \mathbb{N}: n>x$. The negation is $\exists x\in \mathbb{R}\forall n\in\mathbb{N}:n\leq x$, right?
From the Archimedean Axiom we have that $\forall X\in \mathbb{R} \exists n\in \mathbb{N}: n>X$. At the negation we have that for this $n$ there exists a $Y\in\mathbb{R}$ such that $n\leq Y$.
So, we have that $X<n\leq Y$, right?
At the proof we have: There also exists $a,b\in \mathbb{Z}$ such that $a<xn$ and $b>yn$, where $n\in \mathbb{N}$ and $x,y\in \mathbb{R}.
From $X<n\leq Y$ we also get $-Y<-n\leq -X$, therefore an integer is between two reals, right?
So, since $a,b\in\mathbb{Z}$, we have that $a<xn$ and $b>yn$, where $xn, yn\in\mathbb{R}$.
[Some summary of the recent finding in Mathworks: Topic 1]
Predicative mathematics have injections and surjections linked to different things: injection measures size while surjection measures complexity. In ZF type set theories however, both injection and surjection are tied to size, which becomes cardinality for well ordered sets
Also it is consistent with predicative mathematics that the universal object is countable
For example, the Hartogs number proof of $\omega_1$ relies on that the collection of countable ordinals is a set, but that requires the powerset axiom, which assumes @user21820 (can you help me to fill in the details?)
@TobiasKildetoft, Can you please provide an example for some metric space $X$, a point $p\in X$ and a positive real number $r$ such that all three of the following different: a) open ball centered at $p$ with radius $r$, call it $N_r(p)$, (b) closure of $N_r(p)$ and (c) closed ball centered at $p$ with radius $r$
Do you mean the following? We have that $\forall x\in \mathbb{R} \exists n\in\mathbb{N}: n>x$. For $x=-y>0$, $\exists n\in \mathbb{N}: n>-y \Rightarrow y>-n$.
@LeakyNun I had a look at your solutions. Looks good in general. A few places where you seem to go into unnecessary detail (such as apparently proving the isomorphism theorem rather than just using it), and a few where the level of detail seemed slightly too low, but not by much
Good morning! Which math problems are you wrestling today, channel?
I am trying to morph functions to other functions smoothly. But I have something of a bug which makes my solution hoppy. Maybe I just forgot some cost term in my optimization, I am not sure.
Mathematics is the study of formal objects and rules, whose properties and truth value can be unambiguously determined. Using this we can define the following:
well there exists modal logic and fuzzy logic, thus truth values can be probabilistic
(and we have beyesian and frequentist interpretations)
Also experimental mathematics and applied mathematics count because we do have a formal system plus rules that suits the applied or algorithmic situations
i digged this "Godel Escher Bach" book when I was younger also although i never studied much music theory so I probably did not understand all the bachy things, although i love listening to music.
Ah, I am really bad at infinities. Do you mean like cardinalities in set theory?
However, it seems in predicative mathematics, we don't even have uncountability, though complexity and cardinality corresponds to surjectivity and injectivity
I probably lack too much of the foundations of set theory to be speculating too much about it. I like programming pretty images. Maybe I know like the most basic algebra to investigate some things about it. But not higher level stuff.
Let $(x_n)$ be a sequence in some topological space $X$. Initially, I suspected that if $A := \{x_n ~|~ n \in \Bbb{N} \}$ is not closed, then $(x_n)$ must be a convergent sequence. But I think this is wrong. Let $(x_n)$ be some enumeration of the rational numbers; surely such a sequence does not converge. But $A = \Bbb{Q}$ is not closed in $\Bbb{R}$. Does this sound right?
(May not be correct stuff) Given a formal object A, a subobject B is defined to be one which inherits the properties of A. If B derives A, then B is stronger than A.
A formal object and rules (usually a formal system) is said to be a foundation of mathematics if it derives mathematics itself. Therefore a submathematics is a subobject of mathematics. For example, constructive mathematics is a submathematics of mathrmatics
(Note that the term "submathematics" does not really exists in the literature, thus all of this is rambles)
