Sets and the Number Line

In mathematics, a set is a group of things. The things do not have to be tangible objects; they can be abstract things. Each of the things in a set is called an element of the set. In algebra, the elements of a set are often numbers; in geometry they are often points, which are infinitesimally small locations in space. The number of elements in a set could be countable or could be infinite, as long as the elements are quantifiable, definable, or determinable in some way either now or in the future. Infinity is not a number, it is a concept - it means that however big of a number you can think of, infinity is bigger. The objective of algebra is often to take the information available in a situation or problem, and to define or determine a set of elements such as numbers as simply and precisely as possible. There may be no elements in a set; such a set is called an empty set or a null set. If all the elements of one set are also elements of a second set, then the first set is a subset of the second set. In algebra, letters symbolizing sets are commonly capital (upper case) letters, whereas variables standing for numbers are often symbolized by small (lower case) letters. Elements of a series of closely related variables are sometimes symbolized by a letter followed by a subscript number (integer) such as x₁, x₂, etc. The elements in such a set can be more generally symbolized by the letter followed by a subscript lower case letter such as i, j, k, etc. standing for the subscript numbers; for example, x_i where i could stand for 1, 2, 3, etc.

A set can be symbolized with braces around a list of symbols representing the elements of the set, with each element being separated by a comma. For example, a set S containing natural or whole numbers from 1 to 8, inclusive, could be shown as follows:

An empty set is symbolized as follows: { } or the symbol ∅. Real numbers can be represented on a number line, a line theoretically extending infinitely in two opposite directions as shown here:

The arrowheads at the opposite ends of the drawing of the number line mean that line in concept extends infinitely in those directions, even though the drawing of the line cannot be extended forever in those directions. Note that the right side of the number line stretches to positive infinity and the left side stretches to negative infinity. Numbers in a set can be shown as dots on (or near) a number line. For example, the above set of natural numbers from 1 to 8 would be shown as follows: