Relation (mathematics)

In mathematics, a relation denotes some kind of relationship between two objects in a set, which may or may not hold.

As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa.

For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to".

[2][10] The statement (x,y) ∈ R reads "x is R-related to y" and is written in infix notation as xRy.

[7][8] The order of the elements is important; if x ≠ y then yRx can be true or false independently of xRy.

The representation of Rdiv as a Boolean matrix is shown in the middle table; the representation both as a Hasse diagram and as a directed graph is shown in the left picture.

Again, the previous 3 alternatives are far from being exhaustive; as an example over the natural numbers, the relation xRy defined by x > 2 is neither symmetric (e.g. 5R1, but not 1R5) nor antisymmetric (e.g. 6R4, but also 4R6), let alone asymmetric.

The best-known examples are functions[f] with distinct domains and ranges, such as sqrt : N → R+.

Illustration of an example relation on a set A = { a, b, c, d } . An arrow from x to y indicates that the relation holds between x and y . The relation is represented by the set { (a,a), (a,b), (a,d), (b,a), (b,d), (c,b), (d,c), (d,d) } of ordered pairs.
The representation of the relation R el = { ( x , y ) ∈ R × R | x 2 + xy + y 2 = 1 } as a 2D-plot yields an ellipse .
Representation R div as Hasse diagram (black lines) and directed graph (all lines)
Examples of four types of relations over the real numbers : one-to-one (in green), one-to-many (in blue), many-to-one (in red), many-to-many (in black). 2D-plot representation is used.