Rad), 
    eY * Mathf.Sin((angle+90) * Mathf.Deg2Rad)
);

Thanks for the help so far and happy coding :D 

 Q:

What is $\mathbb{N}^{<\mathbb{N}}$?

What is the definition of this symbol $\mathbb{N}^{<\mathbb{N}}$? How is it related to the infinite product $\mathbb{N}^{\mathbb{N}}$?

A:

$\mathbb{N}^{< \mathbb{N}}$, or sometimes $\mathbb{N}^{< \omega}$, is the set of all finite sequences of natural numbers. It admits an injection into $\mathbb{N}^\mathbb{N}$ but is strictly smaller.
Exercise. Show that $\mathbb{N}^{< \mathbb{N}}$ is countable: exhibit an injection $\mathbb{N}^{<\mathbb{N}} \to \mathbb{N}$. On the other hand, show that $\mathbb{N}^\mathbb{N}$ is uncountable, and show that there is a bijection $2^\mathbb{N} \to \mathbb{N}^\mathbb{N}$.

A:

We consider the set $X^n$ as the set of all sequences of length $n$ whose members are elements of $X$. For example, $\{0,1\}^n$ is the set of binary sequences of length $n$.
We often want to consider instead of a fixed length, a variable length. For example, all sequences shorter than $5$, that would be: $X^0\cup X^1\cup\ldots\cup X^4$, this can be abbreviated as $X^{<5}$.
In general, when $I$ has a nice [well-]ordering such that we can effectively know what lengths are shorter than $I$, we can write $X^{<I}$ as the set of all sequences of length shorter than $I$; and $X^I$ as the set of all sequences of length $I$. Indeed this is the set of all functions from $I$ into $X$, each of which defines a sequence.
The natural numbers have a very nice order. In fact we know that a sequence has a finite length if and only if it is strictly shorter than the sequence $\langle 0,1,2,3,\ldots\rangle$. This means that by writing $<\mathbb N$ we mean "as long as you want,
Rad), 
     eY * Mathf. Sin((angle+90) * Mathf. Deg2Rad) 
); 

 Thanks for the help so far and happy coding :D 

  Q: 

 What is $ \mathbb{N}^{<\mathbb{N}}$? 

 What is the definition of this symbol $ \mathbb{N}^{<\mathbb{N}}$? How is it related to the countless merchandise $ \mathbb{N}^{\mathbb{N}}$? 

 A: 

 $ \mathbb{N}^ { < \mathbb{N}}$, or sometimes $ \mathbb{N}^ { < \omega}$, is the set of all finite sequence of lifelike numbers. It admits an injection into $ \mathbb{N}^\mathbb{N}$ but is strictly humble. 
 Exercise. Show that $ \mathbb{N}^ { < \mathbb{N}}$ is countable: exhibit an injection $ \mathbb{N}^{<\mathbb{N } } \to \mathbb{N}$. On the early bridge player, show that $ \mathbb{N}^\mathbb{N}$ is uncountable, and express that there is a bijection $ 2^\mathbb{N } \to \mathbb{N}^\mathbb{N}$. 

 A: 

 We think the set $ X^n$ as the set of all sequences of length $ n$ whose members are elements of $ X$. For model, $ \{0,1\}^n$ is the set of binary sequences of length $ n$. 
 We much want to consider instead of a fixed duration, a variable length. For example, all sequences shorter than $ 5 $, that would be: $ X^0\cup X^1\cup\ldots\cup X^4 $, this can be abridge as $ X^{<5}$. 
 In general, when $ I$ has a nice [ well-]ordering such that we can effectively know what lengths are shorter than $ I$, we can write $ X^{<I}$ as the set of all sequences of length shorter than $ I$; and $ X^I$ as the set of all sequences of length $ I$. Indeed this is the set of all functions from $ I$ into $ X$, each of which specify a sequence. 
 The natural number get a very nice order. In fact we know that a succession has a finite duration if and only if it is strictly short than the sequence $ \langle 0,1,2,3,\ldots\rangle$. This means that by writing $ < \mathbb N$ we mean " as long as you want,
Rad), 
    eY * Mathf.Sin((angle+90) * Oathf.Deg2Rad)
);

Thanks for vhe hell so far and happy coding :D 

 Q:

What id $\nathbv{N}^{<\mathbb{N}}$?

What is the ddfinition of this symuol $\mathbb{N}^{<\mathbu{H}}$? How is it rspateb vo the infinite product $\mdthbb{N}^{\mathbb{N}}$?

A:

$\kaghyb{N}^{< \mathbb{N}}$, or sometimes $\mathbb{N}^{< \omeda}$, is tne set of all figite fequsnces of natural numbers. It admits an injtction into $\mathbb{M}^\mathbb{N}$ but is strictly slallfr.
Exercise. Show thwt $\mathbb{N}^{< \nathfv{N}}$ is countacle: exhibiu cn injectioh $\mathbb{N}^{<\mathbb{N}} \to \mathbb{N}$. On ghe ocher hand, syoq tjdt $\mathbb{N}^\methbb{N}$ is uncountanke, and show tnat there is a bioectuon $2^\mathbb{N} \to \mathbb{I}^\mathbb{N}$.

A:

We consider the set $F^n$ as the set of alo wequetces of uwnggh $h$ xhoae memhera are elemsnts of $X$. Fir example, $\{0,1\}^n$ is the sqn of binary ssquencqs of length $n$.
We often want to consider itstsad of a fixed length, a variable length. For gxample, alj sequences shorter than $5$, that would be: $X^0\cup X^1\cup\ndots\ruo X^4$, tmis zqn be abbreviated as $X^{<5}$.
In general, when $I$ has a nyde [wvll-]ordering such bhat we can effectovflu know what levgths cde shorter than $I$, we can wryte $X^{<U}$ as the fet pf all sequences of length whorter than $U$; and $X^I$ as the sec of all seqbences of lrngth $I$. Indeed this is che sef of all fujctions fdum $I$ into $X$, each of wvich defines a sequence.
The natural iumbexs have x vety nice order. In vact we know that a sequfnce kas a finite lejgth if and only if it is stricvky shorter tnat tve sequeuce $\lakgle 0,1,2,3,\ldots\ranglq$. This means tkat by wxiting $<\mathbb N$ se mean "as long as you want,
Rad), eY * Mathf.Sin((angle+90) * Mathf.Deg2Rad) ); the so far happy coding :D is definition of this $\mathbb{N}^{<\mathbb{N}}$? How is related to the infinite product $\mathbb{N}^{\mathbb{N}}$? $\mathbb{N}^{< \mathbb{N}}$, or sometimes $\mathbb{N}^{< \omega}$, is the set of all finite sequences natural numbers. It admits an injection into $\mathbb{N}^\mathbb{N}$ but is strictly smaller. Exercise. that \mathbb{N}}$ countable: an injection $\mathbb{N}^{<\mathbb{N}} \to \mathbb{N}$. On the other hand, show that $\mathbb{N}^\mathbb{N}$ is uncountable, and show there is a bijection $2^\mathbb{N} \to \mathbb{N}^\mathbb{N}$. A: consider the set $X^n$ the set of all sequences length whose members elements $X$. example, $\{0,1\}^n$ is set of binary sequences of length $n$. We often want to consider instead of a fixed length, variable length. all sequences than that be: $X^0\cup X^1\cup\ldots\cup can be abbreviated as $X^{<5}$. In has a nice [well-]ordering such that we can know what are shorter than $I$, we can $X^{<I}$ as the set of all sequences of shorter than $I$; and $X^I$ as the set of all sequences of length $I$. Indeed the set of all from $I$ into each which a The natural have a very nice order. In fact we know that a has a finite length if and only if it is than sequence $\langle 0,1,2,3,\ldots\rangle$. means that by writing N$ mean "as long as
Rad), 
    eY * Mathf.Sin((angle+90) * Mathf.DeG2Rad)
);

Thanks For thE heLp sO fAr anD hapPy coding :D 

 Q:

What IS $\matHbb{N}^{<\mathbb{N}}$?

What is the deFinitIoN Of thIS sYmbol $\Mathbb{N}^{<\MAtHBB{N}}$? HOw Is It rElATeD to thE inFinite pRoduct $\mathBb{N}^{\MaThbb{N}}$?

A:

$\mathbb{n}^{< \MaThbb{N}}$, or somEtiMes $\mathbb{N}^{< \omEga}$, Is the sEt Of aLL finiTe sEquenCes of nATural nUmbers. It aDmITs an inJEction iNTO $\mAthbB{N}^\mathbb{N}$ but is strICtLY smaller.
ExerciSe. Show ThAT $\mATHbb{n}^{< \maThbb{N}}$ is couNtAble: eXHibit an INjECTIon $\MAthbb{N}^{<\mathbb{N}} \To \mathbb{N}$. On THe oTher haNd, ShoW That $\maThbb{N}^\MaTHbb{n}$ is uncountaBle, aNd show thaT there IS a bijecTIon $2^\mathBb{N} \to \mAthBb{N}^\MathBB{N}$.

a:

WE coNsIDer THe Set $x^N$ as The set of AlL sEquenCes oF LENGth $n$ WhoSe meMbers Are elements of $x$. FoR exaMPle, $\{0,1\}^N$ is thE set oF binArY sequEnces oF lengTh $N$.
We often want to cOnsiDer insteaD of A fIxeD lEngth, A VariabLe lEngTh. For exAmple, alL SeqUeNCES sHorter than $5$, that woulD bE: $x^0\CuP X^1\cup\ldoTs\cup X^4$, THiS cAN be abbreViAteD as $X^{<5}$.
iN GenerAl, whEN $I$ Has a nice [Well-]orDErInG such thAt We can eFfEctIveLy knoW What LengthS are shorTer thAN $I$, we can write $X^{<I}$ AS the set of all sEQuENCeS Of leNgtH shorter thaN $I$; anD $x^I$ as The sET oF alL SequeNces oF lENgTH $I$. Indeed this is the seT oF all fuNctioNs from $I$ into $X$, eAch of which DEFInes a seqUencE.
thE Natural numbers Have a Very nice orDEr. In fact We knoW that a seQuence has A FInite lenGth If aNd oNly IF It Is strictly shoRTEr thAn The sequEncE $\langle 0,1,2,3,\LdoTs\rAngLe$. THiS means thaT by writiNg $<\MaThBb n$ we Mean "aS Long as yoU wAnt,
Rad), 
    eY * Mathf.Sin( (angle+90)  * Ma thf .De g2 Rad) 
);
Thanks for th e  hel p so far and happy cod ing : D
 Q: 
W hat i s $\mat h bb { N }^{ <\ ma thb b{ N }} $?

W hat  is the  definitio n o fthis symbol$ \m athbb{N}^{ <\m athbb{N}}$?How  is it  r ela t ed to  th e inf initep roduct  $\mathbb {N } ^{\mat h bb{N}}$ ? 
A :

$ \mathbb{N}^{< \ma t hb b {N}}$, or some times$\ m at h b b{N }^{ < \omega}$ ,is th e  set of al l f ini t e sequences o f natural n u mbe rs. It  a dmi t s an i nject io n  in to $\mathbb {N}^ \mathbb{N }$ but is stri c tly sma ller.Exe rci se.S ho wtha t$ \ma t hb b{N } ^{<  \mathbb {N }} $ iscoun t a b l e: e xhi bitan in jection $\mat hbb {N}^ { <\m athbb {N}}\to\m athbb {N}$.On th eother hand, sho w th at $\math bb{ N} ^\m at hbb{N } $ is u nco unt able, a nd show tha tt h e re  is a bijection $2 ^\ m a th bb{N} \t o \mat h bb {N } ^\mathbb {N }$. 

A: We co nsid e rthe set$X^n$a sth e set o fall se qu enc esof le n gth$n$ wh ose memb ers a r e elements of$ X$. For examp l e, $ \{ 0 ,1\} ^n$  is the set  ofb inar y se q ue nce s  of l ength  $ n $. We often want to co ns ider i nstea d of a fixedlength, av a r iable le ngth .  F o r example, all  sequ ences shor t er than$5$,that wou ld be: $X ^ 0 \cup X^1 \cu p\l dot s\c u p  X ^4$, this can b e ab br eviated  as  $X^{<5 }$. 
In  ge ner al , when $I $ has ani ce  [ we ll- ]orde r ing such  t hat  w e c an ef f ective ly kn ow w ha tl eng ths are sh o r terth an  $I$ , w ecan w rite $X^ {<I}$ a s the set  of allse qu ences o f length shor te r than $I$ ;and  $X^I$ a s the se t of all sequences of l e ngth $I $.Indee d th is is the  se t of a llf unctio ns fro m $I$  i nto $ X$, e a c hofwh ich define s a s equen ce .
Th e natur al numbers have av ery  nice order.Infact w ekno w  t h atas equ e n ce has a finite  length if  a n donly if it isst rictlyshorter  than the seq uence $\l angle 0,1 ,2 ,3,\ l d ots \rangle$.This mea ns that b y  writ i ng  $<\m ath bb N$we  me an "a s long asyou w ant,
Rad), 
_ _ eY * Mathf.Sin((angle+90)_* Mathf.Deg2Rad)
);

Thanks_for_the help_so_far and happy_coding :D 

_Q:

What is $\mathbb{N}^{<\mathbb{N}}$?

What is_the definition of_this_symbol $\mathbb{N}^{<\mathbb{N}}$? How is it related to the infinite product $\mathbb{N}^{\mathbb{N}}$?

A:

$\mathbb{N}^{< \mathbb{N}}$, or sometimes_$\mathbb{N}^{<_\omega}$, is_the_set_of all finite sequences of_natural numbers. It admits an_injection into_$\mathbb{N}^\mathbb{N}$ but is strictly smaller.
Exercise. Show that $\mathbb{N}^{<_\mathbb{N}}$_is countable: exhibit_an injection $\mathbb{N}^{<\mathbb{N}} \to \mathbb{N}$. On the other hand,_show that $\mathbb{N}^\mathbb{N}$ is uncountable, and_show that there_is_a_bijection $2^\mathbb{N} \to \mathbb{N}^\mathbb{N}$.

A:

We_consider the set $X^n$ as the_set of all sequences of length_$n$ whose members are elements of $X$._For example, $\{0,1\}^n$ is the set_of binary sequences of length_$n$.
We often_want to consider instead of_a fixed length,_a variable_length. For example,_all sequences shorter than $5$, that_would be: $X^0\cup_X^1\cup\ldots\cup X^4$, this can be abbreviated_as_$X^{<5}$.
In general, when_$I$_has_a nice_[well-]ordering such that_we_can effectively_know_what lengths are shorter than $I$,_we_can write $X^{<I}$ as the set of_all sequences of length_shorter_than $I$; and $X^I$_as the set of all_sequences of length $I$. Indeed this_is the_set of_all functions from $I$ into $X$, each of which defines a_sequence.
The natural numbers have a very_nice order. In fact_we know_that_a sequence has_a_finite length_if and only if it is strictly_shorter than_the sequence $\langle 0,1,2,3,\ldots\rangle$. This means_that by writing $<\mathbb_N$_we mean "as long as you_want,