Complete Product

The complete products of a Boolean algebra of subsets generated by a set {A_k}_(k=1)^p of cardinal number p are the 2^p Boolean functions

 B_1B_2...B_p=B_1 intersection B_2 intersection ... intersection B_p,

where each B_k may equal A_k or its complement A^__k. For example, the 2^3=8 complete products of A={A_1,A_2,A_3} are


Each Boolean function has a unique representation (up to order) as a union of complete products. For example,

A_1A_2 union A^__3=(A_1A_2A_3 union A_1A_2A^__3) union (A_1A_2A^__3 union A^__1A_2A^__3 union A_1A^__2A^__3 union A^__1A^__2A^__3)
=A_1A_2A_3 union +a_1A_2A^__3 union A^__1A_2A^__3 union A_1A^__2A^__3 union A^__1A^__2A^__3

(Comtet 1974, p. 186).

See also

Boolean Function, Conjunction

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Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 186, 1974.

Referenced on Wolfram|Alpha

Complete Product

Cite this as:

Weisstein, Eric W. "Complete Product." From MathWorld--A Wolfram Web Resource.

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