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GeezerVison is showing mathworld.wolfram.com — Complement Set -- from Wolfram MathWorld

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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^'=⌀,
(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

1=P(S)=P(E union E^').
(4)

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

1=P(E)+P(E^')
(5)
P(E^')=1-P(E).
(6)

This demonstrates that

P(S^')=P(⌀)=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)

See also

Intersection, Poretsky's Law, Set Difference, Symmetric Difference, Universal Set

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References

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

Referenced on Wolfram|Alpha

Complement Set

Cite this as:

Weisstein, Eric W. "Complement Set." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplementSet.html

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