In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ...

For explanation of the symbols used in this article, refer to the table of mathematical symbols. The following table lists many specialized symbols commonly used in mathematics. ...

Basic definition

The intersection of A and B

The intersection of A and B is written "A ∩ B". Formally:

x is an element of A ∩ B if and only if

• x is an element of A and
• x is an element of B.

For example:

• The intersection of the sets {1, 2, 3} and {2, 3, 4} is {2, 3}.
• The number 9 is not 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, …}.

If the intersection of two sets A and B is empty, that is they have no elements in common, then they are said to be disjoint, denoted: A ∩ B = Ø. For example the sets {1, 2} and {3, 4} are disjoint, written
{1, 2} ∩ {3, 4} = Ø. ↔ ⇔ ≡ logical symbols representing iff. ... AND Logic Gate In logic and mathematics, logical conjunction (usual symbol and) is a two-place logical operation that results in a value of true if both of its operands are true, otherwise a value of false. ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In mathematics, any integer (whole number) is either even or odd. ...

More generally, one can take the intersection of several sets at once. The intersection of A, B, C, and D, for example, is A ∩ B ∩ C ∩ D = A ∩ (B ∩ (C ∩ D)). Intersection is an associative operation; thus,
A ∩ (B ∩ C) = (A ∩ B) ∩ C. In mathematics, associativity is a property that a binary operation can have. ...

Arbitrary intersections

The most general notion is the intersection of an arbitrary nonempty collection of sets. If M is a nonempty set whose elements are themselves sets, then x is an element of the intersection of M if and only if for every element A of M, x is an element of A. In symbols: The empty set is the set containing no elements. ... IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation... In predicate logic, universal quantification is an attempt to formalize the notion that something (a logical predicate) is true for everything, or every relevant thing. ...

This idea subsumes the above paragraphs, in that for example, A ∩ B ∩ C is the intersection of the collection {A,B,C}.

The notation for this last concept can vary considerably. Set theorists will sometimes write "∩M", while others will instead write "∩_A∈M A". The latter notation can be generalized to "∩_i∈I A_i", which refers to the intersection of the collection {A_i : i ∈ I}. Here I is a nonempty set, and A_i is a set for every i in I. Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...

In the case that the index set I is the set of natural numbers, you might see notation analogous to that of an infinite series: In mathematics, an index set is another name for a function domain. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, a series is a sum of a sequence of terms. ...

When formatting is difficult, this can also be written "A₁ ∩ A₂ ∩ A₃ ∩ ...", even though strictly speaking, A₁ ∩ (A₂ ∩ (A₃ ∩ ... makes no sense. (This last example, an intersection of countably many sets, is actually very common; for an example see the article on σ-algebras.) In mathematics, a σ-algebra (pronounced sigma-algebra) or σ-field over a set X is a collection Σ of subsets of X that is closed under countable set operations; σ-algebras are mainly used in order to define measures on X. The concept is important in mathematical analysis and probability theory. ...

Finally, let us note that whenever the symbol "∩" is placed before other symbols instead of between them, it should be of a larger size (⋂).

Nullary intersection

Note that in the previous section we excluded the case where M was the empty set (∅). The reason is as follows. The intersection of the collection M is defined as the set (see set-builder notation) The empty set is the set containing no elements. ... In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by indicating the properties that its members must satisfy. ...

If M is empty there are no sets A in M, so the question becomes "which x's satisfy the stated condition?" The answer seems to be every possible x. When M is empty the condition given above is an example of a vacuous truth. So the intersection of the empty family should be "the power set". Vacuous truth is a special topic of first-order logic. ...

A partial fix for this problem can be found if we agree to restrict our attention to subsets of a fixed set U called the universe. In this case the intersection of a family of subsets of U can be defined as In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...

Now if M is empty there is no problem. The intersection is just the entire universe U, which is a well-defined set by assumption.

See also

• Complement
• Intersection graph
• Logical conjunction
• Naive set theory
• Symmetric difference
• Union