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# Poretsky's Law

The theorem in set theory and logic that for all sets and ,

 (1)

where denotes complement set of and is the empty set. The set is depicted in the above Venn diagram and clearly coincides with iff is empty.

The corresponding theorem in a Boolean algebra states that for all elements of ,

 (2)

The version of Poretsky's Law for logic can be derived from (2) using the rules of propositional calculus, namely for all propositions and ,

 (3)

where "is equivalent to" means having the same truth table. In fact, in the following table, the values in the second and in the third column coincide if and only if the value in the first column is 0.

 ( and not ) or (not and ) 0 0 0 0 1 1 1 0 1 1 1 0

This entry contributed by Margherita Barile

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## References

Hall, F. M. An Introduction to Abstract Algebra, Vol. 1, 2nd ed. Cambridge, England: Cambridge University Press, p. 50, 1972.Hall, F. M. An Introduction to Abstract Algebra, Vol. 2, 2nd ed. Cambridge, England: Cambridge University Press, p. 348, 1972.

Poretsky's Law

## Cite this as:

Barile, Margherita. "Poretsky's Law." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PoretskysLaw.html