## Sets

 DeMorgan's Laws. Let $A,B$ and $C$ be sets. Show that $\left(A\cup B\right)\cup C=A\cup \left(B\cup C\right)$, $A\cup B=B\cup A$, $A\cup \varnothing =A$, $\left(A\cap B\right)\cap C=A\cap \left(B\cap C\right)$, $A\cap B=B\cap A$, and $A\cap \left(B\cup C\right)=\left(A\cap B\right)\cup \left(A\cap C\right)$. 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.

## Notes and References

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].

## References

[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.