User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Links to Definition:Indexing Set/Family of Subsets

Let $\left\{ { E_n }\right\}_{n \mathop \in \N}$ be a countably infinite collection of at least two (distinct) sets.

Then there exists a countable infinite collection of disjoint sets $\left\{ { E_n }\right\}_{n \mathop \in \N}$ satisfying:


 * $\displaystyle \bigsqcup_{n \mathop \in \N} F_n = \bigcup_{n \mathop \in \N} E_n$

Proof
We construct such a set of sets.

Define:


 * $\displaystyle F_k = E_k \setminus \left({ \bigcup_{j \mathop = 0}^{k \mathop - 1} E_j }\right)$


 * $E = \bigcup_{j \mathop \in \N} E_j$

for $k \in \N$.

We first show that the union of all $F_k$ is equal to the union of all $E_k$:

Eventually
User:GFauxPas/Sandbox/Zeta2/lnxln1-x/existence

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/integrand

User:GFauxPas/Sandbox/Zeta2/lnxln1-x/evaluation

User:GFauxPas/Sandbox/Zeta2/FourierSeries/

User:GFauxPas/Sandbox/Zeta2/Informal Proof

User:GFauxPas/Sandbox/NumberTheory