In sample space 𝑆, for events 𝐴 and 𝐵, what is the set of outcomes for the event 𝐴 union 𝐵 bar?
We begin by recalling what we mean by the notation in the question. 𝐵 bar, also sometimes written 𝐵 prime, is the complement of event 𝐵. This is the set of outcomes not contained in event 𝐵. In this question, we are trying to find the set of outcomes for the event 𝐴 union 𝐵 bar. The union of two events is the set of outcomes that occur in one event or the other or in both of them. From the Venn diagram, the set of outcomes in event 𝐴 are three, six, nine, 12, and 15. These are the numbers in the three times table between one and 15 inclusive. Set 𝐵 contains the three outcomes four, eight, and 12. These are the numbers in the four times table between one and 15.
The complements of event 𝐵 will therefore contain the outcomes that are not in event 𝐵. These are the numbers not in the four times table in the sample space. They are one, two, three, five, six, seven, nine, 10, 11, 13, 14, and 15. We want the union of event 𝐴 and the complement of 𝐵. As the outcomes three, six, nine, and 15 occur in both events, we do not need to count these twice. The union of event 𝐴 and the complement of 𝐵 contains the outcomes one, two, three, five, six, seven, nine, 10, 11, 12, 13, 14, and 15. This is all the outcomes in the sample space except four and eight, as these two outcomes only occur in set 𝐵. The required area on the Venn diagram is everything outside of this section.