Difference between revisions of "Notes:Infinity notation"

Latest revision as of 15:23, 21 October 2016

I think I have made a mistake with the notation:

Suppose we write $\bigcup_{n=1}^\infty$ , if we have $\bigcup_{n=1}^\infty A_n$ for a sequence $({ A_n })_{ n = 1 }^{ \infty }$ all is well, from the expression we can tell it means the union of all terms in the sequence.

However take:

∞⋃n=1Xn, where Xn is to be interpreted as all n-tuples of elements of X
- Does this mean all finite tuples, or does it include $X^\mathbb{N}$ ?

Typically when we write $\bigcup_a^b$ we mean starting at $a$ and proceeding towards $b$ in the obvious way, and including $b$ , for example:

$\bigcup_{i=1}^5 A_i$ is $A_1\cup A_2\cup A_3\cup A_4\cup A_5$ , so when we encounter an $\infty$ (which in this case... if anything means $\aleph_0$ ) we should attempt to include it!

The solution currently being considered is:

⋃n∈NAn, this has the advantage of:
- $\left[x\in\bigcup_{n\in\mathbb{N} }A_n\right]\iff\left[\exists n\in\mathbb{N}(x\in A_n)\right]$ (by definition of union), this is exactly what we mean when we write this.

What about ∑∞n=1an? Should we write ∑n∈Nan instead? This also has ∑5i=1ai being the sum from a1 to a5 inclusive.
- This is sidestepped by saying:
  - $\sum^\infty_{n=1}a_n$ is an expression/notation/syntatic sugar for writing $\lim_{n\rightarrow\infty}\left(\sum_{k=1}^na_k\right)$
- Of course also we cannot sum infinite terms, nor is there an $a_\infty$ term in a sequence. We can only sum finitely many times (in a ring, or group)

This page is some notes on a solution to this problem, and to mention "irregularities" that may result.

@@ Line 1: / Line 1: @@
 ==Overview==
-I think I have made a mistake, with the notation:
+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}} where {{MSeq|A_n}} is a [[sequence]] all is well, from the expression we can tell it means the union of all terms in the sequence. But take:
+* 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.
-** {{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}}
+However take:
-*** Does this mean all ''finite'' tuples, or does it include {{M|X^\mathbb{N} }}?
+* {{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:
 * {{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!