Can you explain to me how to get the numbers. Thank you

Total 40 people.

28 like jogging
19 like swimming


Smallest possible n(J∩S) means smallest possible number who likes both. So we should have as few people in this category.

If we have more people who like both, the number of people who like neither has to be higher in order to achieve the total of 40 people.

So we want to have 0 people who likes neither.


28 + 19 = 47
47 - 40 = 7

We have too many people. Over by 7 people. This means we have double counted 7 people.

This means that 7 people like both swmming and jogging, hence the double count.

If we had wanted the biggest n(J∩S) instead, it would be 19.

We get S be a subset of J. Which means that all 19 people who like swimming also like jogging.


28 - 19 = 9
So 9 people like jogging only, and 19 like both.

40 - 28 = 12

We can say that 12 people do not like either activity

Alternative for a)

n(J∪S) = n(J) + n(S) - n(J∩S)


Assuming that there is no one who likes neither activity, (every one likes at least 1 activity)


n(J∪S) = 40

So 40 = 28 + 19 - n(J∩S)

n(J∩S) = 47 - 40 = 7

b) largest n(J∪S) = 40

This means that everyone likes at least 1 activity.

There isn't anybody that likes neither.


But since question asks for n(J∪S)', this means the largest number of people who like neither.

This is done by making S a subset of J, as previously mentioned in the earlier comment. All those who like swmming also likes jogging.

Which means that 28 people like jogging, out of which 9 like jogging only and 19 like both jogging and swimming.


Max number of people who like neither
= 40 - 28
= 12

Edited.