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Union

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The union of two sets A and B is the set obtained by combining the members of each. This is written A union B, and is pronounced "A union B" or "A cup B." The union of sets A_1 through A_n is written union _(i=1)^nA_i. The union of a list may be computed in the Wolfram Language as Union[l].

Let A, B, C, ... be sets, and let P(S) denote the probability of S. Then

P(A union B)=P(A)+P(B)-P(A intersection B).
(1)

Similarly,

P(A union B union C)=P[A union (B union C)]
(2)
=P(A)+P(B union C)-P[A intersection (B union C)]
(3)
=P(A)+[P(B)+P(C)-P(B intersection C)]-P[(A intersection B) union (A intersection C)]
(4)
=P(A)+P(B)+P(C)-P(B intersection C)-{P(A intersection B)+P(A intersection C)-P[(A intersection B) intersection (A intersection C)]}
(5)
=P(A)+P(B)+P(C)-P(A intersection B)-P(A intersection C)-P(B intersection C)+P(A intersection B intersection C).
(6)

The general statement of this property for n sets is known as the inclusion-exclusion principle.

If A and B are disjoint sets, then by definition P(A intersection B)=0, so

P(A union B)=P(A)+P(B).
(7)

Continuing, for a set of n disjoint elements E_1, E_2, ..., E_n

P( union _(i=1)^nE_i)=sum_(i=1)^nP(E_i),
(8)

which is the countable additivity probability axiom. Now let

E_i=A intersection B_i,
(9)

then

P( union _(i=1)^nE intersection B_i)=sum_(i=1)^nP(E intersection B_i).
(10)

See also

Disjoint Union, Inclusion-Exclusion Principle, Intersection, OR, Union-Closed Set, Unsorted Union Explore this topic in the MathWorld classroom

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Cite this as:

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

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