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Complete set of Boolean operators

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This is an old revision of this page, as edited by Ruud Koot (talk | contribs) at 16:10, 11 October 2005 (Complete Boolean algebra (computer science) moved to Completeness (Boolean algebra)). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.

In computer science, a complete Boolean algebra is a collection of Boolean operators that permits the realisation of any possible truth table.

Example truth table (Xor):

a b Result
0 0 0
0 1 1
1 0 1
1 1 0

Using a complete Boolean algebra which does not include XOR (such as the well-known AND OR NOT set), this function can be realised as follows:

(a or b) and not (a and b).

However, other complete Boolean algebras are possible, such as NAND or NOR (either gate can form a complete Boolean algebra by itself - the proof is detailed on their pages).

Note that just because a set of gates forms a complete Boolean algebra does not mean that the resulting expressions will be simple. To gain an XOR function using only NAND gates, for example, is a fairly complex expression - the important thing is that it exists.