Problem Sheet - Sets

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 2 July 2011


DeMorgan's Laws. Let A,B and C be sets. Show that
  1. (AB)C =A(BC),
  2. AB = BA,
  3. A=A,
  4. (AB)C =A(BC),
  5. AB = BA, and
  6. A(BC) = (AB) (AC) .
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 Bn 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.

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