Question

the results of a survey of common allergies was organized into a venn diagram.
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?

Explanation:

Step1: Identify the region for "not allergic to any"

The universal set \( U \) has a region outside all three circles (D, C, P) labeled 22. This represents people not allergic to dogs, cats, or pollen.

Step2: Identify the region for "all three"

The intersection of all three circles (D, C, P) is labeled 1. This is the number of people allergic to all three.

Step3: Identify the region for "dogs and cats but not pollen"

The intersection of D and C, excluding the part that also includes P, is labeled 7. This is people allergic to dogs and cats but not pollen.

Step4: Identify the region for "cats only"

The part of circle C that doesn't overlap with D or P is labeled 18. This is people allergic to cats only.

Answer:

• How many people are not allergic to any of the three choices? 22
• How many people are allergic to all three choices? 1
• How many people are allergic to both dogs and cats but not allergic to pollen? 7
• How many people are allergic to cats only? 18