1.3.7.3 Complement of a Set

The complement of a set , denoted as oris a set that contains all the elements that belong to the universal set  but not to . This can be formally represented as:

        

 is the set of all elements  in the universal set  such that the element  is not in set .

Example 1.21. Let's say we have a universal set , containing all English letters {, , , ..., } and a set , containing vowels {, , , , }, represented by Fig 1.8. As a result, the complement of set  would be the set of consonants, represented as: = {, , ..., }.

        

The complement of a set has two important properties, also known as complement laws:

• The union of a set  and its complement , is equal to the universal set , represented as:

        

                

• The intersection of a set  and its complement , is equal to the empty set , represented as:

        

        

Building upon the English letters example, the union of  and  is the set of letters that are either in , or in , which makes it a universal  set itself, i.e.:

        

Represented as:

        

Thus:

        

Furthermore, the intersection of A and  is the set of letters that are in both  and , which makes it equal to the empty set, as there aren’t any letters that belong to both sets (None of the English letters can simultaneously be classified as both a vowel and a consonant.). This relationship can be represented as:

        

Represented as:

        

Thus:

        

Now that we have established a solid foundation in sets, in the upcoming sections, we'll explore how probability makes use of sets to quantify uncertainty and model real-world phenomena.