properties of cardinal numbers

Theorem. Let $\displaystyle\left(c_{\gamma}\right)_{\gamma\in\Gamma}$ be an indexed family of cardinal numbers indexed by a nonempty index set $\Gamma$ . Also, let $\displaystyle\left({\Gamma}_{\delta}\right)_{\delta\in\Delta}$ be an arbitrary indexed partition of the index set. Then we have the following properties:

1. Associative Laws.

$\sum_{\gamma\in\Gamma}{c_{\gamma}}=\sum_{\delta\in\Delta}\sum_{\gamma\in{% \Gamma}_{\delta}}{c_{\gamma}}$

and

$\prod_{\gamma\in\Gamma}{c_{\gamma}}=\prod_{\delta\in\Delta}\prod_{\gamma\in{% \Gamma}_{\delta}}{c_{\gamma}}.$

2. Commutative Laws. Let $\displaystyle\pi:\Gamma\to\Gamma$ be a partition. Then

$\sum_{\gamma\in\Gamma}{c_{\gamma}}=\sum_{\gamma\in{\Gamma}}{c_{\pi(\gamma)}}$

and

$\prod_{\gamma\in\Gamma}{c_{\gamma}}=\prod_{\gamma\in{\Gamma}}{c_{\pi(\gamma)}}.$

3. Distributive Laws. Let $a$ be any arbitrary infinite cardinal number. Then

$a\left({\sum_{\gamma\in\Gamma}{c_{\gamma}}}\right)=\sum_{\gamma\in\Gamma}{ac_{% \gamma}}$

Title	properties of cardinal numbers
Canonical name	PropertiesOfCardinalNumbers
Date of creation	2013-03-22 16:08:29
Last modified on	2013-03-22 16:08:29
Owner	gilbert_51126 (14238)
Last modified by	gilbert_51126 (14238)
Numerical id	7
Author	gilbert_51126 (14238)
Entry type	Theorem
Classification	msc 03-00