Difference between revisions of "Notes:Infinity notation"
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==Overview== | ==Overview== | ||
| − | I think I have made a mistake | + | I think I have made a mistake with the notation: |
| − | * {{M|1=\bigcup_{n=1}^\infty}}, if we have {{M|1=\bigcup_{n=1}^\infty A_n}} | + | * Suppose we write {{M|1=\bigcup_{n=1}^\infty}}, if we have {{M|1=\bigcup_{n=1}^\infty A_n}} for a [[sequence]] {{MSeq|A_n}} all is well, from the expression we can tell it means the union of all terms in the sequence. |
| − | + | However take: | |
| − | + | * {{MM|1=\bigcup_{n=1}^\infty X^n}}, where {{M|X^n}} is to be interpreted as all {{M|n}}-[[tuple|tuples]] of elements of {{M|X}} | |
| + | ** Does this mean all ''finite'' tuples, or does it include {{M|X^\mathbb{N} }}? | ||
Typically when we write {{M|\bigcup_a^b}} we mean starting at {{M|a}} and proceeding towards {{M|b}} in the obvious way, and including {{M|b}}, for example: | Typically when we write {{M|\bigcup_a^b}} we mean starting at {{M|a}} and proceeding towards {{M|b}} in the obvious way, and including {{M|b}}, for example: | ||
* {{M|1=\bigcup_{i=1}^5 A_i}} is {{M|A_1\cup A_2\cup A_3\cup A_4\cup A_5}}, so when we encounter an {{M|\infty}} (which in this case... if anything means [[aleph 0|{{M|\aleph_0}}]]) we should attempt to include it! | * {{M|1=\bigcup_{i=1}^5 A_i}} is {{M|A_1\cup A_2\cup A_3\cup A_4\cup A_5}}, so when we encounter an {{M|\infty}} (which in this case... if anything means [[aleph 0|{{M|\aleph_0}}]]) we should attempt to include it! | ||
Latest revision as of 15:23, 21 October 2016
Overview
I think I have made a mistake with the notation:
- Suppose we write [ilmath]\bigcup_{n=1}^\infty[/ilmath], if we have [ilmath]\bigcup_{n=1}^\infty A_n[/ilmath] for a sequence [ilmath] ({ A_n })_{ n = 1 }^{ \infty } [/ilmath] all is well, from the expression we can tell it means the union of all terms in the sequence.
However take:
- [math]\bigcup_{n=1}^\infty X^n[/math], where [ilmath]X^n[/ilmath] is to be interpreted as all [ilmath]n[/ilmath]-tuples of elements of [ilmath]X[/ilmath]
- Does this mean all finite tuples, or does it include [ilmath]X^\mathbb{N} [/ilmath]?
Typically when we write [ilmath]\bigcup_a^b[/ilmath] we mean starting at [ilmath]a[/ilmath] and proceeding towards [ilmath]b[/ilmath] in the obvious way, and including [ilmath]b[/ilmath], for example:
- [ilmath]\bigcup_{i=1}^5 A_i[/ilmath] is [ilmath]A_1\cup A_2\cup A_3\cup A_4\cup A_5[/ilmath], so when we encounter an [ilmath]\infty[/ilmath] (which in this case... if anything means [ilmath]\aleph_0[/ilmath]) we should attempt to include it!
Possible solution
The solution currently being considered is:
- [ilmath]\bigcup_{n\in\mathbb{N} } A_n[/ilmath], this has the advantage of:
- [ilmath]\left[x\in\bigcup_{n\in\mathbb{N} }A_n\right]\iff\left[\exists n\in\mathbb{N}(x\in A_n)\right][/ilmath] (by definition of union), this is exactly what we mean when we write this.
Counterpoints
- What about [ilmath]\sum^\infty_{n=1}a_n[/ilmath]? Should we write [ilmath]\sum_{n\in\mathbb{N} }a_n[/ilmath] instead? This also has [ilmath]\sum_{i=1}^5 a_i[/ilmath] being the sum from [ilmath]a_1[/ilmath] to [ilmath]a_5[/ilmath] inclusive.
- This is sidestepped by saying:
- [math]\sum^\infty_{n=1}a_n[/math] is an expression/notation/syntatic sugar for writing [math]\lim_{n\rightarrow\infty}\left(\sum_{k=1}^na_k\right)[/math]
- Of course also we cannot sum infinite terms, nor is there an [ilmath]a_\infty[/ilmath] term in a sequence. We can only sum finitely many times (in a ring, or group)
- This is sidestepped by saying:
This page is some notes on a solution to this problem, and to mention "irregularities" that may result.
Practical problems
- A lot of pages use [ilmath]\bigcup_{n=1}^\infty[/ilmath]
