Describing a Set

A set can be described two ways—by roster method or by using set-builder notation.

1. Roster Method

   The roster method simply lists all the elements in the set. For example, set A could be described using braces like this:

   A = {1, 2, 3, 4, 5}.

   We could easily tell what is in the set by just looking at it. The above is a finite set, meaning you can count the number of objects in the set.

   Another version of the roster method that is used when dealing with infinite set is sometimes called the modified roster method because it uses an ellipsis, " … ", to show that it follows a given pattern to infinity. For example, B = {1, 2, 3, $\dots$ } would be the infinite set of all positive integers.

   The modified roster method can also be used on finite sets where there are many terms that follow a pattern. For example, C = {1, 2, 3, $\dots$ 100} would be the set of positive integers from one to one hundred. Here, the three dots show that the numbers follow the pattern set up by the first three numbers to one hundred.

   Note: In set B above, we cannot list all the numbers and we cannot count them, so set B is an infinite set.

   If a set is described using the roster method, you must use braces to enclose the elements.

2. Set-Builder Notation

   The second method of describing a set is set-builder notation, where a set is described using the following format:

   A = {x | x is an even integer > 0}

   It would be read as "A is the set of all x such that x is an even integer greater than zero." This set would look like the following using the roster method:

   A = {2, 4, 6, $\dots$ }.

   The set-builder notation method can also be used to describe a set that would be cumbersome using the roster method. For example,

   A = { x | x is a city in the U.S. that has more than 5,000 people living in it}.

   You can imagine how many cities would be in this set!

Empty Set

Now that we know what a set is and can describe it, we need to talk about a very special set called the empty set or the null set. The more common name is empty set. It is defined as the set having no elements and is denoted by the symbol $\varnothing$ or by empty braces {}.