Set Definitions

A set is a well-defined collection of distinct objects.

First Example Set

There are two main methods for describing sets. We can list all of the elements of a set as we did in the example set containing a triangle, a circle, and a square with a simple diagram. On the other hand, we can describe a set like this: the set of integers greater than 5 and less than 10.

Set Notation

For notation, we will typically enclose the elements of our sets or the descriptions in braces for clarity. So, the set of integers greater than 5 and less than 10 could be written as { integers greater than 5 and less than 10 } or { 6, 7, 8, 9 }. However, we often use this more formal notation { x | x is an integer and 5 < x < 100}, which is read as the set of all x such that x is an integer and x greater than 5 and less than 10; the vertical bar is a read as such that and this is called set-builder notation.

Definition of "Element Of" Symbol

In order to more easily refer to sets, we will usually assign them names like this: A = { 1, 2, 7 }. We use the symbol ∈ to say that an object belongs to a set. The symbol ∈ is reads as is an element of. For example, we write 2 ∈ A to mean 2 is an element of the set A. Likewise, we write 3 ∉ A to mean 3 is not an element of the set A. That covers most of the basic notation.

Examples of Non-Sets

We defined a set as a well-defined collection of distinct objects, and we must be sure that our sets meet these requirements. As an example, the collection { 1, 2, 2, 3 } fails to be a set because the objects are not distinct. Also, the collection { the two largest integers } fails to be a set because it is not well-defined, since there is no such thing as the two largest integers.

Special Sets

There are a few special sets that should be mentioned. The first is the empty set, which looks like this ∅ and represents the set with nothing in it. Then we have sets with a single element in them, called singleton sets. Singleton sets, like this one { 1 } which contains the number 1, are different than the single elements by themselves. Finally, we have the common numerical sets that are used in mathematics shown below:

Number Systems


ℕ - Natural Numbers
ℤ - Integers
ℚ - Rational Numbers
ℝ - Real Numbers
ℂ - Complex Numbers

Finite and Infinite Sets

Our first example sets contained a finite number of elements. The Number System sets differ from our previous examples because the are all infinite sets. We will frequently make use of ellipses to specify infinite sets. For example, we could specify the Natural Numbers as { 1, 2, 3, ... }.