Complement Set

Given a set  with a subset , the complement (denoted  or ) of  with respect to  is defined as

(1)

Using set difference notation, the complement is defined by

(2)

If , then

(3)

where  is the empty set. The complement is implemented in the Wolfram Language as Complement[l, l1, ...].

Given a single set, the second probability axiom gives

(4)

Using the fact that ,

(5)

(6)

This demonstrates that

(7)

Given two sets,

			(8)
			(9)

REFERENCES:

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 2, 1991.