Video Transcript
In the sample space 𝑆, for events
𝐴 and 𝐵, what is the set of outcomes for the event 𝐴 intersection 𝐵?
The notation in our question
denotes the intersection. And we know that the intersection
of events 𝐴 and 𝐵 consists of all outcomes that are in both 𝐴 and 𝐵. We can do this directly from the
Venn diagram by listing the elements in the section where events 𝐴 and 𝐵
overlap. There is only one element here, the
number 12. So we can therefore conclude that
the set of outcomes for the event 𝐴 intersection 𝐵 contains the single element
12.
An alternative way to approach this
problem would be to consider sets 𝐴 and 𝐵 separately first. Set 𝐴 contains the numbers three,
six, nine, 12, and 15. These are the numbers in the three
times table, or 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 one and 15. As we are trying to find the
intersection, we need the elements that are in set 𝐴 and set 𝐵. There is only one such element, the
number 12. This confirms that 𝐴 intersection
𝐵 is the set containing the single element 12. 12 is the only number between one
and 15 inclusive that is a multiple of three and a multiple of four.
Whilst it is not required for this
question, it is worth noting that the numbers one, two, five, seven, 10, 11, 13, and
14 are not multiples of three and not multiples of four. As a result, they are found outside
of the circles on the Venn diagram. In conclusion, we can work out the
elements in any intersection like this directly from the Venn diagram or by listing
the elements in both sets and finding their common elements.
