Interval

Intervals are used extensively in mathematics and are included in every definition. An interval is a set of points that is bounded by two extreme points. We also distinguish between open and closed intervals.

Basic description

Intervals are also used in common speech, for example "you can be fined between ten and twenty thousand crowns by the European Union for the wrong colour of your eyes". In the example, the word 'range' is used, but mathematically it is an interval. But now the tricky questions arise: can you be fined exactly ten thousand crowns, or does the extreme value of the range not count there? How finely can I choose the fine? Can I fine 18,541.97 crowns? Or can I only scale by hundreds, for example? I.e. either 18,500 or 18,600?

The mathematical notation solves everything. There are two important things to say: what set are we in (in the context of the example - hundreds? Thousands? Units of crowns?) and whether the interval includes the extreme points (can I fine just ten or twenty thousand?).

The good news is that in the vast majority of cases in mathematics, we want the interval to work with the set of real numbers, so if you encounter an interval without a set specification, it is certainly real numbers.

We already have the initial set, now how do we indicate if we want to consider the extreme points? Intervals are written using parentheses. A round bracket indicates that the extreme point does not belong in the interval, a sharp bracket indicates that the extreme point belongs there. This is called openness and closedness. If the extreme point belongs to the interval, the interval is closed, if it does not, it is open.

Examples

The following are examples (all in the set of real numbers, unless otherwise stated):

• $\left<1,2\right>\qquad$ A closed interval from one to two. All real numbers between one and two, including one and two, fall into the interval.
• $(1,2)\qquad$ Open interval from one to two. The interval includes all real numbers between one and two, but does not include the numbers one and two themselves.
• $\left<0,1\right)\qquad$ The interval is closed on the left and open on the right. The interval includes all numbers between zero and one, including zero itself, but one does not belong in the interval.
• $\left(p,q\right>\qquad$ The interval is open on the left and closed on the right. All numbers between p and q, including q, but excluding p, fall within the interval.
• $\left<0,\infty\right)\qquad$ The interval is closed on the left and open on the right. If you have an infinity in the interval, use the open interval from that side, the infinity does not have an extreme end point, the closed interval does not make sense there.
• $\left<1,5\right>\subset\mathbb{N}\qquad$ Here follows a change, we are not working with the set of real numbers, but with the set of natural numbers. We have a closed interval on both sides and so there are five numbers in the interval: 1, 2, 3, 4 and 5.
• $\left(1,5\right>\subset\mathbb{N}\qquad$ The same case as a moment ago, only the interval is open from the left and so 1 is not in the interval and the list of all elements of the interval is: 2, 3, 4 and 5.

Interval as a set

As can be seen, an interval is nothing but a subset of the set over which it is defined. Therefore, we can work with intervals as sets to perform set operations on them. For example, a union might look like this:

$$\left<0, 5\right>\cup\left<5{,}10\right>=\left<0{,}10\right>$$

Watch out for open intervals, because there the equality would not hold, because the five does not belong to the resulting interval:

$$\left(0, 5\right)\cup\left(5{,}10\right)\ne\left(0{,}10\right)$$

Correctly, it could look like this:

$$\left(0, 5\right)\cup\left(5{,}10\right)=\left(0{,}10\right)-{5}$$

When to use an open interval

The difference between an open and a closed interval must be understood. A common case of using an open interval is, for example, the definitional domain of the logarithm. We can call the logarithm with any positive real number, which means we can't call it with a negative number, but we can't call it with zero either. How do we write it?

$$D(\ln)=\left(0,\infty\right)$$

This notation says that we can call the logarithm with any small positive number, but we can't call it with zero. So, for example, the number 1 falls in the interval, so does the number 0.001, so does the number 0,0000000001 or 10⁻⁶⁶⁶. But zero no longer falls in the interval. Similarly, we could write the defining range of a function with a fraction, like the ordinary f(x) = 1/x.

$$D(f)=(-\infty,0)\cup(0,\infty)$$

This gives us the set of real numbers without zero. In neither interval is zero in the place of the closed interval, so there will be no zero in the resulting union. However, we can use any small positive number or any large negative number (for negative numbers, -0.001 is greater than -0.1, so any large number and not any small number).

Geometric interpretation

Intervals are usually represented on the number axis as line segments, with the extreme points chosen according to whether the interval is closed or open. If the interval is closed, the circle representing the point will be coloured, if it is open, it will be uncoloured, just a circle.

The following figure shows a representation of three intervals always from two to six, but differing in the closedness of the sides. So, in order, the following intervals will be displayed: <2,6>, (2,6>, (2,6).