Algebra: Simple Sets

Simple Sets

In this Algebra video, we introduce a few of the most fundamental aspects of mathematics and set theory. This lesson covers some of the terminology and concepts about sets that we will use in future algebra lessons. Below, we show some of the common ways we will represent sets.

The first set is the set of letters G, D, and Q and is given by the pictorial diagram and is more concrete than our other representations. The second shows the five integers 3, 56, 34, 6, and 38 inside of braces. Notice that these elements are not in order, even though we often order our lists for clarity. Like the second set, the third set is given in braces, but is more descriptive; this set is just {1,2,3,4}. The fourth and final set is read as "the set of elements x such that x is an integer and x is between 3 and 8." This set is just {4,5,6,7}, but it is given in formal notation that we will commonly use.

Although our first sets above are all finite, we will often use infinite sets like the set of integers. Also, we will refer to the set with no elements in it: this set is called the empty set and is denoted by a circle with a line through it. Sets with a single element like this, {2} are called singleton sets. Below, we show the infinite set integers, the empty set and a singleton set containing 7, respectively.

There is some additional notation that we will use. First, we will typically assign sets a letter name so that we can refer to them more succinctly. Second, we will use the character that looks like a strange "e" to specify that something "is an element of" a given set. We will use the same symbol with a strike through to denote the something "is not an element of" a given set.

Above, we have assigned the set {1,2,4} the name A. The second statement reads "1 is an element of A." The third statement reads "3 is not an element of A."

Furthermore, we define three set-wise operations that we will apply to pairs of sets: Intersection, Union, and Difference. Examples of these are given below.