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

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.