Cardinality != Density?

I remained in a conversation where I said that the thickness of 2 collections of the very same cardinality can be various about the boundless series of non - adverse integers. Does cardinality indicate that any kind of set of ${\aleph_0}$ has equivalent thickness to any kind of various other set of the very same ${\aleph_0}$?

Additionally, does cardinality indicate equivalence in set dimension? Or is an official position of various rates which can not be contrasted?

Ok, obtained that until now. Increasing inquiry currently:

Densities can be various yet the matter coincides. Boundless = = Infinite iff cardinality is equal. Why does not a various thickness indicate boundless set! = boundless set also if cardinality is equal?

2022-06-07 14:41:04
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Answers: 3

Subsets of the naturals can have various natural densties. If the all-natural thickness is more than absolutely no, the part is countably boundless. Consider all the naturals (thickness 1) and also the even numbers (thickness 1/2). Yet there are boundless parts with all-natural thickness absolutely no and also boundless parts for which the all-natural thickness can not be specified.

2022-06-07 15:10:25

Let $\mathbb{N}$ describe the favorable integers. Generally we specify the thickness of a part $A\subset \mathbb{N}$ relative to the integers to be $$\lim_{N\rightarrow\infty}\frac{|\{m\in A:m\leq N\}|}{N}.$$ For instance, the set of also numbers $$\{2,4,6,\dots\}$$ has thickness $\frac{1}{2}$. From this, it is not tough to see that for any kind of $c\in[0,1]$ you can locate a set with thickness $c$.

Notification that if $c>0$ this instantly indicates the set has cardinality $\aleph_0$, yet it is additionally feasible to have a set of thickness $0$ with cardinality $\aleph_0$. As an example, the set of powers of $2$, or the set of prime numbers.

Hope that aids,

2022-06-07 15:10:01

I assume that it is not entirely clear to you what cardinality and also thickness methods.

To start with cardinality is the "rawest" idea of dimension. For 2 collections to have the very same cardinality they require just a bijection, which is to claim "There exists a table with 2 columns, one for each and every set, and also all the components show up specifically as soon as because column."

The idea which you define is called Dedekind infinite , which is to have the very same cardinality as a correct part.

When it comes to thickness, this can be dealt with in numerous facets:

  • A dense ordered set is a set such that for every single $x,y$ there is $z$ which is purely in between them. The rationals are such set, the favorable integers are not.¢ If there is a first and/or last factor (as an example all the non - adverse rationals which are much less or equivalent to one) after that it is normal to exclude completion factors from the thickness standards and also claim that there is a first/last factor too
  • Topological dense sets: if $A$ is a topological room after that $D\subseteq A$ is thick in $A$ if and also just if its junction with every open set is nonempty. The rationals are thick in the actual numbers, in this feeling, given that every non vacant period has a sensible factor inside it

If the topological room is gifted with a straight order geography (as an example the actual numbers) after that a thick embed in the topological feeling is a thick order too (although not the other way around, take into consideration all the actual numbers versus the rationals in between $-1$ to $1$)

The actual numbers offer instance of a large room (its cardinality is continuum) with a really tiny (countable) part which is thick (topologically).

The 3rd idea of dimension is measure , which about converts to quantity. A huge set is a set of favorable action (or equivalent action to the action of the whole room).

The Lebesgue measure is a means to establish the quantity of parts of the actual numbers in the means we desire it to function, that is if we simply change around the set it will certainly not transform its quantity and also if we extend it the quantity will certainly increase as the variable we extended by.

One can construct a set which is of Lebesgue action $0$ (i.e. has no quantity in all), not thick at any kind of factor (that is if a factor is outside the set after that it has an open period which does not converge this set) and also yet it is of the cardinality of the continuum. This can be generalised to have any kind of quantity that we desire too.

Consequently the ideas are rarely relevant, if we have a really tiny (in cardinality) set which is thick (huge topologically) and also an additional set which is large in cardinality yet topologically talking is really really tiny.

To summarize, there are several means to gauge just how large a set is and also it obtains an increasing number of intricate with every new strategy you acquire (filters, cofinality, and also extra). They might or might not relate or associated, yet the instance is generally that we desire a new means of "sizing" up collections, primarily due to the fact that the ones we have want or difficult for the job handy.

2022-06-07 15:09:29