Venn Diagrams and Cardinal Numbers
P URPOSE : The purpose of this activity is to develop skills in the use of Venn diagrams and the Cardinal Number Formula.
R EQUISITES : This activity assumes that students are familiar with the use of Venn diagrams to show set operations and that students understand the Cardinal Number
Formula.
Cardinal Number Formula:
For any two sets A and B n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
D
IRECTIONS
: Draw an appropriate Venn diagram and use the given information along with the Cardinal Number Formula to fill in the number of elements in each region.
Example: n(U) = 43, n(A) = 25, n(A ∩ B) = 5, n(B') = 30
For two sets A and B, the Venn diagram consists of four regions:
Since n(A ∩ B) = 5, then region 3 has 5 elements.
Since n(A) = 25, then region 2 has 25-5=20 elements.
Since n(B') = 30, then region 1 has 30-20=10 elements.
Now, since the Universal set consists of all four regions, then region 4 has 43 – (10 + 20 +
5) = 8 elements. Hence, the final Venn diagram is:

Complete the following for any two sets A and B.
1.
n(A) = 8, n(B) = 14, n(A ∩ B) = 5, n(U) = 23
2.
3.
n(A) = 19, n(B) = 13, n(A ∪ B) = 25, n(A ′ ) = 11 n(B) = 20, n(A ∩ B) = 16, n(A ∪ B) = 30, n(A ∩ B) ′ = 17
4.
n(A ′ ) = 25, n(B) = 28, n(A ′ ∩ B ′ ) = 40, n(A ∩ B) = 10
5.
n(U) = 40, n(B ′ ) = 19, n(A ∪ B) ′ = 8, n(A ∩ B) = 6
For any three sets X, Y, and Z, the Venn diagram consists of eight regions:
Use what you have learned to complete the following for any three sets X, Y, and Z.
6.
n(X) = 57, n(X ∩ Y) = 35, n(X ∩ Z) = 21, n(Y ∩ Z) = 25, n(X ∪ Y) = 81, n(Z) = 49, n(Y) ′ = 52, n(X ∩ Y ∩ Z) = 15
7.
8.
n(X) = 15, n(X ′ ∩ Y ′ ∩ Z ′ ) = 21, n(Y ∩ Z ′ ) = 3, n(X ∩ Y ∩ Z) =5, n(Y ∩ Z) = 13, n(X ∩ Y ′ ) = 9, n(X ∩ Z) = 8, n(Y ∪ Z) = 32 n(X) = 26, n(X ∩ Z ′ ) = 19, n(Y ∩ Z ′ ) = 21, n(X ∩ Y ∩ Z) = 4, n(Z) = 18, n(U) = 56, n(Y ∩ Z) = 6, n(X ∩ Y) = 12.

Venn Diagram Extra Practice n
(
X
) = 57, n
(
X ∩ Y
) = 35, n
(
X ∩ Z
) = 21, n
(
Y ∩ Z
) = 25, n ( X ∪ Y ) = 81, n ( Z ) = 49, n ( Y ) = 52, n ( X ∩ Y ∩ Z ) = 15.
X Y
Z
U n
(
X
) = 15, n
(
X ∩ Y ∩ Z
) = 21, n
(
Y ∩ Z
) = 3, n
(
X ∩ Y ∩ Z
) = 5 n
(
Y ∩ Z
) = 13, n
(
X ∩ Y
) = 9, n
(
X ∩ Z
) = 8, n
(
Y ∪ Z
) = 32,.
X Y
U
Z

Venn Diagram Extra Practice Solution n
(
X
) = 57, n
(
X ∩ Y
) = 35, n
(
X ∩ Z
) = 21, n
(
Y ∩ Z
) = 25, n ( X ∪ Y ) = 81, n ( Z ) = 49, n ( Y ) = 52, n ( X ∩ Y ∩ Z ) = 15.
13
15
X
20
6
15
10
18
Z
Y
14
U n
(
X
) = 15, n
(
X ∩ Y ∩ Z
) = 21, n
(
Y ∩ Z
) = 3, n
(
X ∩ Y ∩ Z
) = 5 n
(
Y ∩ Z
) = 13, n
(
X ∩ Y
) = 9, n
(
X ∩ Z
) = 8, n
(
Y ∪ Z
) = 32,.
21
6
X
1
3
5
13
Z
8
Y
2
U

7. It was once said that Country-Western songs emphasize three basic themes: love, prison, and trucks. A survey of the local Country-Western radio station produced the following data:
12 songs about a truck driver who is in love while in prison
13 about a prisoner in love
28 about a person in love
18 about a truck driver in love
3 about a truck driver in prison who is not in love
2 about people in prison who are not in love and do not drive trucks
8 about people who are out of prison, are not in love, and do not drive trucks
16 about truck drivers who are not in prison d b
T e a c g
L f
P h
U
(a) How many songs were surveyed?
Find the number of songs about:
(b) truck drivers
(c) prisoners
(d) truck drivers in prison
(e) people not in prison
(f) people not in love