Last updates: 2 July 2011
DeMorgan's Laws. Let $A,B$ and $C$ be sets.
Show that
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Let $X$ be a set. Show that the set $S$ of all subsets of $X$ with operations union, intersection and complement is a Boolean algebra. |
These notes are an updated version of notes of Arun Ram from 1994.
A Boolean algebra is a complemented distributive lattice. References for Boolean algebras are [Brk, Chapt X] and [St, &Sect;3.4]. In particular, see the conditions for the finite Boolean algebra ${B}_{n}$ found in [St, p. 107-108].
[Brk] G.D. Birkhoff, Lattice Theory, ?????
[St] R.P. Stanley, Enumerative combinatorics, Vol. 1, ????
[Ram] A. Ram, Notes in abstract algebra, University of Wisconsin, Madison 1993-1994.
[Bou] N. Bourbaki, Algèbre, Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272 Hermann, Paris, 1959, 211 pp. MR0107661.
[Ru] W. Rudin, Real and complex analysis, Third edition, McGraw-Hill, 1987. MR0924157.