# Math Help - Set Theory.

1. ## Set Theory.

Hi, could anyone help me start off this proof:

Notation: For a set $X$ we define $\bigcup\ X \;=\;\{x \mid x\in y\ for\ some\ y\in X \}$

Then show if $X_{ij}$ for $i,j\in\mathbb{N}$ are sets then:

$\bigcap\limits^\infty_{i=0}(\bigcup\limits^\infty_ {j=0}X_{ij})\;=\;\bigcup\{( \bigcap\limits^\infty_{i=0}X_{ih(i)})\mid h\in\mathbb{N^{N}}\}$

2. If $x\in LHS$ then $\left( {\forall i} \right)\left( {\exists j} \right)\left[ {x \in X_{i~j} } \right]$.
We are going to use that to define $h:\mathBB{N}\to\mathBB{N}$ by $h:i\mapsto j$.
So $x\in X_{i~h(i)}$.

3. Thanks Plato, I think understand what you're saying but doesn't this only give $\bigcap\limits^\infty_{i=0}(\bigcup\limits^\infty_ {j=0}X_{ij})\;\subseteq\;\bigcup\{( \bigcap\limits^\infty_{i=0}X_{ih(i)})\mid h\in\mathbb{N^{N}}\}$?
If so, how would I prove $\supseteq$