de Morgan’s laws for sets (proof)
Let be a set with subsets for , where
is an arbitrary index-set. In other words, can be finite,
countable, or uncountable. We first show that
where denotes the complement of .
Let us define
and . To establish the equality , we shall
use a standard argument for proving equalities in set theory. Namely,
we show that and .
For the first claim, suppose is an
element in .
Then , so for any .
Hence for all , and .
Conversely, suppose is an
element in . Then for all .
Hence for any , so ,
and .
The second claim,
follows by applying the first claim to the sets .
Title | de Morgan’s laws for sets (proof) |
---|---|
Canonical name | DeMorgansLawsForSetsproof |
Date of creation | 2013-03-22 13:32:16 |
Last modified on | 2013-03-22 13:32:16 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 7 |
Author | mathcam (2727) |
Entry type | Proof |
Classification | msc 03E30 |