A college conducted a survey to explore the academic interests and achievements of its students. It asked students to place checks beside the numbers of all the statements that were true of them. Statement #1 was “I was on the Dean’s list last term,” statement #2 was I belong to an academic club, such as the math club or the Spanish club.” and statement #3 was b1 am majoring in at least two subjects.” Out of a sample of 100 students, 28 checked #1, 26 checked #2, and 14 checked #3, 8 checked both #1 and #2, 4 checked both #1 and #3, 3 checked both #2 and #3, and 2 checked all three statements.
a. How many students checked at least one of the statements?
b. How many students checked none of the statements?
c. Let H be the set of students who checked #1, C the set of students who checked #2, and D the set of students who checked #3. Fill in the numbers for all eight regions of the diagram.

d. How many students checked #1 and #2 but not #3?
e. How many students checked #2 and #3 but not #1?
f. How many students checked #2 but neither of the other two?

(a)

The number of students who checked at least one of the statements.

Given information:

In a survey to evaluate the academic interest and the achievements of college students, three statements were asked from sample of 100 students.

The statements were as follows.

Statement #1⇒ “I was on the Dean’s list last term.”

Statement #2⇒ “I belong to an academic club.”

Statement #1⇒ “I am majoring in at least two subjects.”

The students in the sample have responded as follows.

28 students checked the statement #1.

26 students checked the statement #2.

14 students checked the statement #3.

8 students checked both statements #1 & #2.

4 students checked both statements #1 & #3.

3 students checked both statements #2 & #3.

2 students checked all three statements.

Formula used:

The inclusion/exclusion rule for three sets.

Let A,B and C be three finite sets.

Then,

N(A∪B∪C)=N(A)+N(B)+N(C)−N(A∩B)−N(A∩C)−N(B∩C)+N(A∩B∩C)

Calculation:

Let A,B and C be three finite setssuch that,

A is the student who checked the statement #1

B is the student who checked the statement #2

C is the student who checked the statement #3

Then we can summarize the given data using the set notation

(b)

The count of the students who have not checked any of the statements.

(c)

To construct a Venn diagram for the given scenario considering three sets H,C and D for the three statements.

(d)

The number of students who have checked #1 and #2 but not #3.

(e)

The number of students who checked only #2 and #3, but not #1.

(f)

The number of students who checked only #2, but not #1 and #3.