# Relations

Up a level : Algebra and Arithmetic
Next page : Classification of relations

Relations

In mathematics one can define a relation as a structure where we associate one or several elements with each other. This we may write as

$R({p_1},{p_2},{p_3},...,{p_n})$

where ${p_1},{p_2},{p_3},...,{p_n}$ is an ordered list of parameters or variables. The number of variables is called the arity of the relation. The relation is said to hold for, or be satisfied by a particular assignment of the variables if it is true for that assignment. The relation will thus be a logical statement about variables.

The relation a>b, where a and b are real numbers is satisfied by, for example a=5 and b=2.344, since it is true for those numbers, whereas the relation a=b is false for those particular numbers.

Other examples of relation is for example x+y+z=12 or that x and y has the same father (e.g. Luke and Leia), or that two cities are connected to the same railway system. For the later R(Stockholm, Uppsala) is true, whilst R(Stockhalm, Reykavik) is false.

Equations

To find values that satisfies a relation is called to solve the relation. Used in this way the relation is called an equation.

We may for example have the equation 2x=10, that is satisfied by (and only by) x=5, and this will thus be the solution The equation x=5+y is satisfied by an infinite number of values for x and y where x is 5 larger than y, for example x=3 and y=8. The complete solution may be easiest expressed by the very equation itself: x=5+y.

Binary Relations

The last relation we looked at is of a a particularly important type of relations, the so called binary relations, that are the relations between two variables. It may, for the two variables a and b, be written as R(a,b), or, aRb.

We may write our last example as

$xRy \Leftrightarrow x = 5 + y$

This means that xRy is true if and only if x=y+5 is true. The relation R will then be the relation x=y+5. We then say that x is R-related to y.

Relations as sets

A binary relation may be described using equations or descriptions in words, but also as sets of ordered pairs. The relation a>b, where a and b are restricted to the numbers 1, 2, and 3 may be represented by the set

{(3, 2), (3, 1), (2 ,1)},

because these three pairs represent the three cases (a,b) where a>b.

We can represent relations with higher arity with sets of ordered lists with more than two elements.

Domain and range

For the moment being we sill restrict this to binary relation. Say we have a binary relation xRy. We may then choose to see one of the variables as an input, and the other as an output. The one we use as an input we call the independent variable, and the output we call the dependent variable, since it is (usually) depending on the chosen value of the other. The domain of the relation is the set of possible values of the independent variable, and the range is (mostly) defined as the set of possible values of the dependent variable. This is also called the image of the domain. (Note that the range is sometimes defined as set that has the image as a subset, but might include other elements too.)

Say, for example ,that the relation is “y is the length of x rounded to the nearest cm”, where the domain is the set of human beings . Then the range is the interval of lengths (in cm) from the smallest human being that have existed to the tallest that have existed.

The domain is thus the set of allowed inputs, and the range the set of possible outputs.

Up a level : Algebra and Arithmetic
Next page : Classification of relationsLast modified: Jan 13, 2017 @ 20:51