Section 2.4: Using Sets to Solve Problems
Using Venn Diagrams with Two Sets to Solve Problems
Step 1 Find the number of elements that are common to both sets and write
that number in region II.
Step 2a Find the number of elements that are in set 𝐴 and not set 𝐵 by
subtracting the number in region II from the total number of elements in 𝐴.
Then write that number in region I.
Step 2b Repeat for the elements in 𝐵 but not in region II, and write in
region III.
Step 3 Find the number of elements in 𝑈 that are not in either 𝐴 or 𝐵, and
write it in region IV.
Step 4 Use the diagram to answer specific questions about the situation.
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EXAMPLE 1 In 2008, there were 36 states that had some form of casino
gambling in the state, 42 states that sold lottery tickets of some kind, and 34
states that had both casinos and lotteries. Draw a Venn diagram to represent
the survey results.
How many states have only casino gambling?
How many states have only lotteries?
How many states have casino gambling and lotteries?
How many states have casino gambling or lotteries (or both)?
How many states have neither?
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EXAMPLE 2 In a survey published in the Journal of the American Academy
of Dermatologists, 500 people were polled by random telephone dialing. Of
these, 120 reported having a tattoo, 72 reported having a body piercing, and
41 had both. Draw a Venn diagram to represent these results.
How many respondents have only tattoos?
How many respondents have only body piercings?
How many respondents have have tattoos and body piercings?
How many respondents have tattoos or body piercings (or both)?
How many respondents have have neither?
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Using Venn Diagrams with Three Sets to Solve Problems
Region I represents the elements in set 𝐴 but not in set 𝐵 or set 𝐶.
Region II represents the elements in set 𝐴 and set 𝐵 but not in set 𝐶.
Region III represents the elements in set 𝐵 but not in set 𝐴 or set 𝐶.
Region IV represents the elements in sets 𝐴 and 𝐶 but not in set 𝐵.
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Region V represents the elements in sets 𝐴, 𝐵, and 𝐶.
Region VI represents the elements in sets 𝐵 and 𝐶 but not in set 𝐴.
Region VII represents the elements in set 𝐶 but not in set 𝐴 or set 𝐵.
Region VIII represents the elements in the universal set 𝑈 , but not in set
𝐴, 𝐵, or 𝐶.
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EXAMPLE 3 A survey of 300 first-year students at a large midwestern university was conducted to aid in scheduling for the following year. Responses
indicated that 194 were taking a math class, 210 were taking an English class,
and 170 were taking a speech course. In addition, 142 were taking both math
and English, 111 were taking both English and speech, 91 were taking both
math and speech, and 45 were taking all three. Draw a Venn diagram to
represent these survey results.
(a) Find the number of students taking Only English.
(b) Find the number of students taking Math and speech but not English.
(c) Find the number of students taking Math or English.
(d) Find the number of students taking none of these three subjects.
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