Math Revision 16.1 - Relation and Function

In these revision notes for Relation and Function, we cover the following key points:

• What is an ordered pair? In what conditions are two ordered pairs equal?
• What is the Cartesian plane? What are the elements in each axis?
• What is the Cartesian product? What is it made from?
• What is the Cartesian square? How it is shown in a Cartesian plane?
• What is a relation in math?
• How many sets are needed to form a relation? What are their names?
• What are the methods used to represent a relation?
• What are the symbols used in the math relations terminology?
• What is a function? Where does it differ from relation?
• How to denote functions?
• How to find the value of a function?
• What is the domain of a function?
• What is the range of a function.

Relation and Function Revision Notes

In mathematics, an ordered pair is a set of two numbers usually written in the form (a, b). These two numbers are taken from different sets, usually determined by values in the two perpendicular number axes - the horizontal and vertical ones. The order of the two numbers is important in the sense that (a, b) is different from (b, a) unless a = b. Ordered pairs are commonly used to specify a location on a coordinate plane.

Ordered pairs are not always connected through lines. In many cases, they are disconnected points in the coordinate system. This occurs when the data set shown on the graph represents the relationship between the variables through scattered points.

By definition, the set composed of all these ordered pairs is called the Cartesian product of the two sets A and B. In symbols, we represent the Cartesian product as A × B.

An example of the application of the Cartesian product in practice is the coordinates system, otherwise known as the Cartesian plane, where all ordered pairs represent a specific point of the coordinates system.

A Cartesian square is obtained when all elements of the same set A multiply each other. In symbols, we express the Cartesian square as A × A or A² (hence the name Cartesian square). The Cartesian plane is actually an example of the Cartesian square because the elements in each direction are numbers, i.e. they belong to the same set - the set of real numbers R. The particular of Cartesian square elements is that they have the same number of inputs and outputs.

If some ordered pairs are connected through specific rules, they form a relation. We represent schematically a relation in math as follows:

In math relations, there may be some output values which have more than one input value in correspondence. The reverse can also occur in relations.

There are a number of ways to describe the relations. Each of these ways (methods) has its own advantages and disadvantages and is more suitable in a particular context. These methods are:

a) Venn Diagram Method

Venn diagrams are tools used to visually describe the similarities, differences and types of connections between the elements of two number sets: input and output. A Venn diagram is a kind of circle or ellipse, where the elements are spread out in the inner part. The corresponding values are connected through arrows.

b) Set Builder Method

In this method of expressing relations, we use mathematical notation (symbols) to describe the combination of elements of the two sets (input X and output Y) according to a pre-defined rule. All conditions required to express the relation are written one after another separated by a comma. Some of the symbols used in this method are:

The symbol "∊" means "is an element of ". It is used to describe situations in which an element (number) belongs to a number set. For example, "3 ∊ A" means "the number 3 is an element of the number set A".

The symbol "⊂" means "is a subset of ". It is used to describe situations in which a smaller set is part (subset) of a larger set of numbers. For example, if A = {3, 6, 9} and B = {0, 3, 6, 9, 12}, we write A ⊂ B, as all elements of A belong to B and the latter has more elements not contained in A.

The symbol "⋀" means "and". It is used when two conditions apply simultaneously to a given situation. For example, if we have A = {natural numbers smaller than 10 ⋀ divisible by 3}, it is obvious that A = {3, 6, 9} because the two above conditions must apply simultaneously.

Curled brackets { } are used to describe number sets.

c) Roster Method (Form)

The roster form is the simplest method for representing relations. In this method, all elements of the relation are listed inside a set of brackets, where all the possible ordered pairs of the two sets that follow the given relation are written.

By definition, a function is a special type of relation, where each element of the independent variables set X has in correspondence a single element of the dependent variables set Y.

In other words, in a function, an x-value has in correspondence a single y-value. The reverse is not a must, i.e. a y-value may have in correspondence more than an x-value.

There are several methods to denote a function. The first is to express the function as an equation with two variables, where the y-variable is the dependent one (as always). For example, we can write y = 2x - 1, y = 3x^x + 2, y = 3/x, etc.

Another method to express functions is to write it as f(x), which we read as "the image of x". This is because in functions, all mathematical objects we choose (x-values) have in correspondence a single image (a y-value), similar to images produced in mirrors. For example, we can write the functions above as f(x) = 2x - 1, f(x) = 3x^x + 2, f(x) = 3/x, etc.

The third method to express functions is as x → f(x). We read this notation as "the values of x are mirrored to those of f(x)". For example, the functions above are written as x → 2x - 1, x → 3x^x + 2, x → 3/x, etc.

Evaluating a function means finding the images (y-values) of certain values of the independent variable (i.e. of x-values) by substituting the values of the independent variable x in the formula of the function. In this way, we find the corresponding y-values, which are the images required.

By definition, the set of the allowed values for the x-variable in a function is known as the domain D. In our example, we have D = [2, 4]. (Recall the symbols of segment and interval explained in chapter 10.)

On the other hand, the set of images resulting from the substitution of the input values from the domain D in the formula of a function is called the range R.