Example 1: A travel agent surveyed 100 people to find out how many of them had visited the cities of Melbourne and Brisbane. Thirty-one people had visited Melbourne, 26 people had been to Brisbane, and 12 people had visited both cities. Draw a Venn diagram to find the number of people who had visited:
a Melbourne or Brisbane
b Brisbane but not Melbourne
c only one of the two cities
d neither city.
Solution
Let M be the set of people who had visited Melbourne, and let B be the set of people who
had visited Brisbane. Let the universal set E be the set of people surveyed.
The information given in the question can now be rewritten as
|M|=31,|B|=26,|MnB|=12 and |E|=100.
Hence number in M only=31–12 =19
and number in B only =26–12 =14.
(a.) Number visiting Melbourne or Brisbane = 19 + 14 + 12 = 45.
(b.) Number visiting Brisbane only = 14.
(c.) Number visiting only one city = 19 + 14 = 33.
(d.)Number visiting neither city = 100 – 45 = 55.