Complement Set (original) (raw)

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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 theWolfram 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,

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." FromMathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ComplementSet.html

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