Problems similar to the problem posed at the beginning of this session can be represented using a Venn diagram, cardinal numbers, and the ideas of union and intersection. We use these concepts to solve the problem.

 A group of 30 students went on a camping trip.  Of these, 12 return with both sunburns and insect bites, and 20 report sunburn.  How many suffered only insect bites if it is known that only three students suffered neither?

 To solve it we construct a Venn diagram and enter the cardinal number for each section.

Solution:  The 12  that had both are in the intersection of sunburn and insect bites.  
The 3 that had neither are in the universe, but outside of the circles.
Since 20 had sunburn and 12 of those also had insect bites, we conclude 8 had only sunburn.
And finally, since there were 30 students total, and we have  3 + 8 + 12 = 23 accounted for, there must be 30 – 23 = 7 who had insect bites only.

        If we let S represent the set of students who had a sunburn and B represent the set of students who suffered insect bites, we would have the following symbolic relationships with their interpretation.

Symbolic                    Interpretation

  n(U) = 30                    Thirty students went on the camping trip.

n(S ∪ B) = 27             Twenty-seven students received a sunburn or an insect bite.

n(S ∩ B) = 12             Twelve students received a sunburn and an insect bite.