2. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
Draw the universal set for the 50 students with two overlapping
circles and label it with the total in each.
Example:
M S
U
3. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
Because 8 students belong to both clubs, put “8” in the overlap.
Example:
M S
8
U
4. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
Account for 8 of the 20 Math Club members, leaving 12 students
who are members of the Math Club but are not members of the
Science Club.
Example:
M S
8
12
U
5. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
Account for 8 of the 34 Science Club members, leaving 26 students
who are members of the Science Club but are not members of the
Math Club.
Example:
M S
8
12 2
6
U
6. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
This shows that a total of 12 + 8 + 26 or 46 students are in either
Math Club or Science Club. This leaves 4 students unaccounted for
so these students must be the ones that are in neither clubs. Thus,
Example:
M S
8
12 2
6
4
U
7. Out of 50 students, 20 are members of the Math Club and 34 are
members of the Science Club. If 8 are in both clubs, how many
students are in a) neither of the clubs and b) either clubs?
Solutio
n:
a. There are 4 students that are in neither of the clubs.
Example:
M S
8
12 2
6
4
U
b. There are 46 students that are in either clubs.
8. Example:
In a certain high school, 100 grade 7 students were asked on what
clubs they joined. Among them, 38 joined the Math Club, 35 joined
the Science Club, 37 joined the English Club, 8 joined both the Math
and Science Clubs, 6 joined both the Math and English Clubs, 6
joined both the Science and English Clubs, and 3 joined all three
clubs.
1. How many students joined Math Club only?
2. How many students joined Math and Science Clubs but not the
English Club?
3. How many students joined the English Club but not Math or
Science Clubs?
4. How many students did not join in any of the three clubs?
9. Solutio
n: Draw a Venn diagram with three overlapping circles representing
the following sets:
M = set of students who joined Math Club
E = set of students who joined the English Club
S = set of students who joined the Science Club
M E
S
10. Solutio
n:
Since three students joined the three clubs, then the cardinality of M ∩
E ∩ S is 3. The Venn diagram will be as shown below. Let U be the
universal set.
M E
S
U
3
11. Solutio
n:
Since six students joined both the Science and English Clubs, there are 6
– 3 = 3 students who joined both Science and English Clubs but not the
Math Club. The Venn diagram will be as shown below.
M E
S
U
3
3
12. Solutio
n:
Since six students joined both the Math and English Clubs, there are 6 – 3
= 3 students who joined both Math and English Clubs but not the
Science Club. The Venn diagram will be as shown below.
M E
S
U
3
3
3
13. Solutio
n:
Following the same procedure, since eight students joined both the Math
and Science Clubs, there are 8 – 3 = 5 students who joined both Math
and Science Clubs but not the English Club. The Venn diagram will be as
shown below.
M E
S
U
3
3
3
5
14. Solutio
n:
Note that there are 38 students who joined the Math Club. It follows that
there are 38 – (5 + 3 + 3) = 38 – 11 = 27 students who joined the Math
club only. The Venn diagram will then be modified as shown below.
M E
S
U
3
3
3
5
2
7
15. Solutio
n:
Also, there are 37 students who have joined the English Club, and hence,
there are 37 – (3 + 3 + 3) = 37 – 9 = 28 students who have joined the
English Club only. The Venn diagram will be as shown below.
M E
S
U
3
3
3
5
2
7
2
8
16. Solutio
n:
There are 35 students who have joined the Science Club, and so, there
are 35 – (5 + 3 + 3) = 35 – 11 = 24 students who joined the Science Club
only. The Venn diagram will then result as follows:
M E
S
U
3
3
3
5
2
7
2
8
2
4
17. Solutio
n:The total number of students who joined any of the three clubs is the
sum of cardinalities inside the three circles in the Venn diagram, that is 27
+ 5 + 3 + 3 + 3 + 28 + 24 = 93. However, there were 100 students and
so, 100 – 93 = 7 students did not join in any of the three clubs. The Venn
diagram will then be as follows:
M E
S
U
3
3
3
5
2
7
2
8
2
4
7
18. Solutio
n:
Looking at the Venn diagram, you can now answer the questions easily.
1. How many students joined the Math Club only?
2. How many students joined Math and Science Clubs but not the English
Club?
3. How many students joined the English Club but not Math or Science
Clubs?
4. How many students did not join any of the three clubs?
M E
S
U
3
3
3
5
2
7
2
8
2
4
7
2
7 5
2
8
7
