In set theory, the intersection of two sets The notion of intersection as an algebraic operation with sets as operands has been generalized from geometry, where it is encountered in the case of geometric sets of points, such as individual points, lines (infinite uncountable sets of points), planes, etc.

Notation and terminology

" between the terms; that is, in infix notation which is similar to capital-sigma notation.

For an explanation of the symbols used in this article, refer to the table of mathematical symbols.

Definition

,

if and only if

For example:

• The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.

• The number 9 is in the intersection of the set of prime numbers {2, 3, 5, 7, 11, ...} and the set of odd numbers {1, 3, 5, 7, 9, 11, ...}, because 9 is not prime.

• The intersection of two geometric sets of points such as two lines is a singleton set of one point for distinct non-parallel lines in the same plane.

Intersecting and disjoint sets

We say that in which case we also say that is an

We say that are disjoint if their intersection is empty

For example:

• are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.

• two parallel lines in the same plane are disjoint.

Algebraic properties

Binary intersection is an associative one has

. Intersection is also commutative The intersection of any set with the empty set Also, the intersection operation is idempotent. All these properties follow from analogous facts about logical conjunction.

Intersection distributes over union one may define the complement may be written as the complement of the union of their complements, derived easily from De Morgan's laws

Arbitrary intersections

The most general notion is the intersection of an arbitrary is a nonempty is an element of the if and only if for every

The notation for this last concept can vary considerably. Set theorists

In the case that the index set is the set of natural numbers, notation analogous to that of an infinite product

". This last example, an intersection of countably many sets, is actually very common; for an example, see the article on σ-algebras.

Nullary intersection

was the empty set is defined as the set (see set-builder notations satisfy the stated condition?" The answer seems to be is empty, the condition given above is an example of a vacuous truth. So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), but in standard (ZF) set theory, the universal set does not exist.

This matches the intuition that as collections of subsets become smaller, their respective intersections become larger; in the extreme case, the empty collection has an intersection equal to the whole underlying set.

Also, in type theory).

See also

References

Further reading

External links