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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,

(1)

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

A_1A_2A_3,A_1A_2A^__3,A_1A^__2A_3,A^__1A_2A_3, A_1A^__2A^__3,A^__1A_2A^__3,A^__1A^__2A_3,A^__1A^__2A^__3.

(2)

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

(Comtet 1974, p. 186).

See also

Boolean Function, Conjunction

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References

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

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Complete Product

Cite this as:

Weisstein, Eric W. "Complete Product." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CompleteProduct.html

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