1.3.1.2 Set Builder Form

In set builder form, the elements of the set are described by a statement or expression. In this method, we do not list the elements; instead, we will write the representative element using a variable (generally written as ) followed by a vertical line ( | ) or colon ( : ) and write the general property of the same representative element. For instance, the same set above (that denotes the set of letters in the word, "probability") can be written as:

        

Or

        

This can be read as “ is the set of all , such that  is a letter of the word probability”.

Example 1.10. Consider a set  of the first five prime numbers. Using roster form  can be represented as:

        

Using set builder form,  can be represented as:

        

This can be read as “ is the set of all , such that  is a prime number and less than or equal to 11”.

The true benefit of employing set builder form becomes apparent when we encounter scenarios where we must represent a set that contains an extensive number of elements, or where it proves challenging to represent the set using the roster form. This notation captures the essence of the set without requiring an exhaustive list of its elements. A few examples representing this notion are:

Examples	Set Representation
Even Numbers	{ |  is an integer and  is divisible by 2}
Positive Integers Less Than 10	{ |  is an integer and 1 ≤  < 10}
Countries in Europe	{ |  is a country located on the continent of Europe}
Customers Who Purchased a Product	{ |  is a customer of a specific store and  purchased a particular product}
Positive Reviews for a Movie	{ |  is a review for a movie and  is positive (e.g., rated above 3 stars)}
Successful Job Applicants	{ |  is a candidate who applied for a job and  was offered the position}