Question

A survey of 91 college students was taken to determine the musical styles they liked. Of those, 34 students listened to techno, 20 to hip hop, and 31 to rap. Also, 17 students listened to techno and rap, 10 to techno and hip hop, and 7 to hip hop and rap. Finally, 5 students listened to all three musical styles. Construct a Venn diagram and determine the cardinality for each region. Use the completed Venn diagram to answer the following questions:

a. How many listened to only techno music?

b. How many listened to hip hop and rap, but not techno?

c. How many listened to hip hop or rap, but not techno?

d. How many listened to music in exactly one of the musical styles?

Answer 1

a. The number of students who listened to only techno music is 12.

b. The number of students who listened to hip hop and rap, but not techno, is 2.

c. The number of students who listened to hip hop or rap, but not techno, is 46.

d. The number of students who listened to music in exactly one of the musical styles is 80.

How to construct a Venn diagram

To construct a Venn diagram and determine the cardinality for each region, start by representing the three musical styles—techno, hip hop, and rap—as three intersecting circles.

Based on the given information, determine the cardinality for each region:

The number of students who listened to only techno music:

To find this number, subtract the students who listened to other combinations from the total number of students who listened to techno music.

Therefore, the number of students who listened to only techno music is:

34 (total students who listened to techno) - 17 (techno and rap) - 10 (techno and hip hop) + 5 (all three styles) = 12.

The number of students who listened to hip hop and rap, but not techno:

To find the number of students who listened to hip hop and rap, but not techno, subtract the number of students who listened to all three styles:

7 (hip hop and rap) - 5 (all three styles) = 2.

The number of students who listened to hip hop or rap, but not techno:

To find this number, add the students who listened to hip hop and rap separately.

From the information given, 20 students listened to hip hop and 31 students listened to rap.

However, this counts the students who listened to all three styles twice. Therefore, subtract the number of students who listened to all three styles once:

20 (hip hop) + 31 (rap) - 5 (all three styles) = 46.

The number of students who listened to music in exactly one of the musical styles:

To find this number, add the students who listened to each style separately and subtract the students who listened to multiple styles (i.e., subtract the students who listened to all three styles twice).

From the information given, 34 students listened to techno, 20 students listened to hip hop, and 31 students listened to rap.

However, subtract the number of students who listened to all three styles

34 (techno) + 20 (hip hop) + 31 (rap) - 5 (all three styles) = 80.