de Morgan’s laws for sets (proof)

Let X be a set with subsets AiX for iI, where I is an arbitrary index-set. In other words, I can be finite, countableMathworldPlanetmath, or uncountable. We first show that

(iIAi) = iIAi,

where A denotes the complementPlanetmathPlanetmath of A.

Let us define S=(iIAi) and T=iIAi. To establish the equality S=T, we shall use a standard argument for proving equalities in set theoryMathworldPlanetmath. Namely, we show that ST and TS. For the first claim, suppose x is an element in S. Then xiIAi, so xAi for any iI. Hence xAi for all iI, and xiIAi=T. Conversely, suppose x is an element in T=iIAi. Then xAi for all iI. Hence xAi for any iI, so xiIAi, and xS.

The second claim,

(iIAi) = iIAi,

follows by applying the first claim to the sets Ai.

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