Question

The results of a survey of common allergies was organized into a Venn diagram.

Circles D, C, and P overlap. Circle D contains 15. Circle C contains 18. Circle P contains 9. The overlap of circles C and D contains 7. The overlap of circles D and P contains 12. The overlap of C and P contains 10. The overlap of all 3 circles contains 1.

Answer the questions about the following sets:

D = {x | x is a person allergic to dogs}; C = {x | x is a person allergic to cats}; P = {x | x is a person allergic to pollen}



How many people are not allergic to any of the three choices?

How many people are allergic to all three choices?

How many people are allergic to both dogs and cats but not allergic to pollen?

How many people are allergic to cats only?

Answer 1

First note: the triple-overlap is given as 1, so "all three" = 1.

Assuming the usual interpretation that pairwise overlaps (e.g. C∩D = 7) include the people counted in all three, we get:

- Allergic to all three: 1.

- Allergic to both dogs and cats but not pollen: |C∩D| − |C∩D∩P| = 7 − 1 = 6.

- Allergic to cats only: |C| − |C∩D| − |C∩P| + |C∩D∩P|
= 18 − 7 − 10 + 1 = 2.

- Number not allergic to any: cannot be determined from the information given, because you need the total number of people surveyed.

(Also note: the given numbers are inconsistent as a whole — inclusion–exclusion gives |D∪C∪P| = 15+18+9 −7−12−10 +1 = 14, which is impossible since it is less than |D| = 15. So at least one of the provided counts must be incorrect.)