Intersection

The intersection (red) of two disks (white and red with black boundaries).

The circle (black) intersects the line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points.

The intersection of D and E is shown in grayish purple. The intersection of A with any of B, C, D, or E is the empty set.

In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them.

Intersections can be thought of either collectively or individually, see Intersection (geometry) for an example of the latter. The definition given above exemplifies the collective view, whereby the intersection operation always results in a well-defined and unique, although possibly empty, set of mathematical objects. In contrast, the individual view focuses on the separate members of this set. Given this view, intersections need not be unique, as shown by the two points of intersection between a circle and a line pictured. Similarly, (individual) intersections need not exist as between two parallel but distinct lines in Euclidean geometry.

Intersection is one of the basic concepts of geometry. An intersection can have various geometric shapes, but a point is the most common in a plane geometry. Incidence geometry defines an intersection (usually, of flats) as an object of lower dimension that is incident to each of the original objects. In this approach, an intersection can be sometimes undefined, such as for parallel lines. In both the cases the concept of intersection relies on logical conjunction. Algebraic geometry defines intersections in its own way with intersection theory.

In set theory

Considering a road to correspond to the set of all its locations, a road intersection (cyan) of two roads (green, blue) corresponds to the intersection of their sets.

The intersection of two sets A and B is the set of elements which are in both A and B. Formally,

$A\cap B=\{x:x\in A\text{ and }x\in B\}$.

For example, if $A=\{1,3,5,7\}$ and $B=\{1,2,4,6\}$, then $A\cap B=\{1\}$. A more elaborate example (involving infinite sets) is:

$A=\{x:\text{ x is an even integer}\}$

$B=\{x:\text{ x is an integer divisible by 3}\}$$\text{ , then}$

$A\cap B=\{6,12,18,\dots \}$

As another example, the number 5 is not contained in the intersection of the set of prime numbers {2, 3, 5, 7, 11, …} and the set of even numbers {2, 4, 6, 8, 10, …} , because although 5 is a prime number, it is not even.

Sets can have an empty intersection. For example, if $A=\{Amy,Aaron,Arjun\}$ and $B=\{Barack,Bas,Betty\}$, then $A\cap B=\mathrm{\varnothing }$. Such sets are called disjoint sets and may colloquially be said to have no intersection.

Notation

Intersection is denoted by the U+2229 ∩ from Unicode Mathematical Operators.

The symbol U+2229 ∩ was first used by Hermann Grassmann in Die Ausdehnungslehre von 1844 as general operation symbol, not specialized for intersection. From there, it was used by Giuseppe Peano (1858–1932) for intersection, in 1888 in Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann.

Peano also created the large symbols for general intersection and union of more than two classes in his 1908 book Formulario mathematico.

See also

• Constructive solid geometry, Boolean Intersection is one of the ways of combining 2D/3D shapes
• Dimensionally Extended 9-Intersection Model
• Meet (lattice theory)
• Intersection (set theory)
• Union (set theory)