Complement Set

Given a set S with a subset E, the complement (denoted E^' or E^_) of E with respect to S is defined as

E^'={F:F in S,F not in E}.

(1)

Using set difference notation, the complement is defined by

E^'=S\E.

(2)

If E=S, then

E^'=S^'=emptyset,

(3)

where emptyset 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

1=P(S)=P(E union E^').

(4)

Using the fact that E intersection E^'=emptyset,

1=P(E)+P(E^')

(5)

P(E^')=1-P(E).

(6)

This demonstrates that

P(S^')=P(emptyset)=1-P(S)=1-1=0.

(7)

Given two sets,

P(E intersection F^') = P(E)-P(E intersection F)

(8)

P(E^' intersection F^') = 1-P(E)-P(F)+P(E intersection F).

(9)

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More things to try:

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