### Video Transcript

In the sample space 𝑆, for events 𝐴 and 𝐵, what is the set of outcomes for the event the complement of 𝐴 intersection the complement of 𝐵.

We begin by recalling the notation in this question. 𝐴 bar and 𝐵 bar denote the complement of 𝐴 and the complement of 𝐵. These are the elements not in set 𝐴 and not in set 𝐵, respectively. The 𝑛 symbol denotes the intersection. We are therefore looking for elements that are not in set 𝐴 and not in set 𝐵. Set 𝐴 contains the numbers three, six, nine, 12, and 15. These are the multiples of three between one and 15 inclusive. Set 𝐵 contains the elements four, eight, and 12. These are the multiples of four between the numbers one and 15 inclusive.

The elements or numbers that are not in set 𝐴 and not in set 𝐵 are those that are outside of our two circles. These are the numbers one, two, five, seven, 10, 11, 13, and 14. These eight numbers are the integers between one and 15 inclusive that are not multiples of three or multiples of four.

An alternative method here would be to list the elements that are in the complement of 𝐴 and the complement of 𝐵. The complement of 𝐴 is the set of numbers that are not in 𝐴. This is equal to one, two, four, five, seven, eight, 10, 11, 13, and 14. In the same way, the complement of 𝐵 consists of all those numbers that are not in set 𝐵. It is all the integers from one to 15 except four, eight, and 12. The elements that are in both of these sets are one, two, five, seven, 10, 11, 13, and 14. This is the set of outcomes for the event the complement of 𝐴 intersection the complement of 𝐵.