In a survey where 100 students reported which subjects they like, 32 students in total liked Mathematics, 38 students liked Business and 30 students liked Literature. Moreover 7 students liked both Mathematics and Literature, 10 students liked both Mathematics and Business, 8 students liked both Business and Literature, 5 students liked all three subjects. Then the number of people who liked exactly one subject is
65
This question involves analyzing survey data about student preferences for three subjects: Mathematics, Business, and Literature. We need to determine how many students liked exactly one of these subjects.
Survey Data Overview
We are given the following information from a survey of 100 students:
- Total students surveyed: 100
- Number of students who liked Mathematics ($|M|$): 32
- Number of students who liked Business ($|B|$): 38
- Number of students who liked Literature ($|L|$): 30
- Number of students who liked Mathematics and Literature ($|M \cap L|$): 7
- Number of students who liked Mathematics and Business ($|M \cap B|$): 10
- Number of students who liked Business and Literature ($|B \cap L|$): 8
- Number of students who liked all three subjects ($|M \cap B \cap L|$): 5
Calculating Subject Overlaps
To find the number of students who liked exactly one subject, we first need to determine the number of students in each specific overlap category using the provided data. We can use the principle of inclusion-exclusion or visualize this with a Venn diagram.
The number of students who liked all three subjects is given:
- Liked all three ($M \cap B \cap L$): 5
Now, let's find the number of students who liked exactly two subjects:
- Liked Mathematics and Business, but not Literature: $|M \cap B| - |M \cap B \cap L| = 10 - 5 = 5$
- Liked Business and Literature, but not Mathematics: $|B \cap L| - |M \cap B \cap L| = 8 - 5 = 3$
- Liked Mathematics and Literature, but not Business: $|M \cap L| - |M \cap B \cap L| = 7 - 5 = 2$
Calculating Exactly One Subject Preference
With the overlap calculations done, we can now find the number of students who liked only one subject. This is done by taking the total number of students who liked a subject and subtracting those who liked it in combination with other subjects.
Mathematics Only Calculation
Number of students who liked only Mathematics:
$|M \text{ only}| = |M| - (|M \cap B \text{ only}| + |M \cap L \text{ only}| + |M \cap B \cap L|)$
$|M \text{ only}| = 32 - (5 + 2 + 5) = 32 - 12 = 20$
Business Only Calculation
Number of students who liked only Business:
$|B \text{ only}| = |B| - (|M \cap B \text{ only}| + |B \cap L \text{ only}| + |M \cap B \cap L|)$
$|B \text{ only}| = 38 - (5 + 3 + 5) = 38 - 13 = 25$
Literature Only Calculation
Number of students who liked only Literature:
$|L \text{ only}| = |L| - (|M \cap L \text{ only}| + |B \cap L \text{ only}| + |M \cap B \cap L|)$
$|L \text{ only}| = 30 - (2 + 3 + 5) = 30 - 10 = 20$
Total Students Liking Exactly One Subject
Finally, to find the total number of students who liked exactly one subject, we sum the numbers calculated for each subject individually:
Total = (Mathematics only) + (Business only) + (Literature only)
Total = $20 + 25 + 20 = 65$
Therefore, 65 students liked exactly one subject.
| Subject Combination | Calculation | Count |
|---|---|---|
| Mathematics & Business & Literature | Given | 5 |
| Mathematics & Business only | $|M \cap B| - |M \cap B \cap L|$ | $10 - 5 = 5$ |
| Business & Literature only | $|B \cap L| - |M \cap B \cap L|$ | $8 - 5 = 3$ |
| Mathematics & Literature only | $|M \cap L| - |M \cap B \cap L|$ | $7 - 5 = 2$ |
| Mathematics only | $|M| - (\text{M&B only}) - (\text{M&L only}) - (\text{All three})$ | $32 - 5 - 2 - 5 = 20$ |
| Business only | $|B| - (\text{M&B only}) - (\text{B&L only}) - (\text{All three})$ | $38 - 5 - 3 - 5 = 25$ |
| Literature only | $|L| - (\text{M&L only}) - (\text{B&L only}) - (\text{All three})$ | $30 - 2 - 3 - 5 = 20$ |
| Total Exactly One Subject | Sum of 'only' counts | $20 + 25 + 20 = 65$ |
Important Questions from Operations on Sets
Match List I with List II
Let R 1= {(1, 1), (2, 2), (3, 3)} and R 2 = {(1, 1), (1, 2), (1, 3), (1, 4)}
List I
List II
(A) R 1∪ R 2
(I) {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (3, 3)}
(B) R 1- R 2
(II) {1, 1}
(C) R 1∩ R 2
(III) {(1, 2), (1, 3), (1, 4)}
(D) R 2- R 1
(IV) {(2, 2), (3, 3)}
Choose the correct answer from the options given below:
Let R be a relation on a set A such that R = R-1, then R is
Which of the following is an open set?
If C = { 2, 4, 6, 8, 10, 12, 14, 16 }, and D = {5, 10, 15, 20}, then the number of elements in the set D - C is:
In a beauty contest, half the number of experts voted for Mr. A and two third voted for Mr. B. 10 voted for both and 6 did not for either. How many experts were there in all?
