Poretsky's Law
The theorem in set theory and logic that for all sets A and B,
| B=(A intersection B^_) union (B intersection A^_)<=>A=emptyset, |
(1)
|
where A^_ denotes complement set of A and emptyset is the empty set. The set (A intersection B^_) union (A^_ intersection B) is depicted in the above Venn diagram and clearly coincides with B iff A is empty.
The corresponding theorem in a Boolean algebra R states that for all elements a,b of R,
| b=(a ^ b^') v (a^' ^ b)<=>a=0. |
(2)
|
The version of Poretsky's Law for logic can be derived from (2) using the rules of propositional calculus, namely for all propositions P and Q,
| Q is equivalent to [(P and not Q) or (not P and Q)] iff P is false, |
(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.
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.Referenced on Wolfram|Alpha
Poretsky's LawCite this as:
Barile, Margherita. "Poretsky's Law." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/PoretskysLaw.html