Venn Diagrams
A Venn Diagram allows us to have a pictoral representation of sets. In a Venn
Diagram we have a universal set U which is represented by the region that
contains everything else. For example, A ⊆ U is
Most of the time the universal set will not be explcitly stated, but we always
need to draw it. Consider the following examples.
Exercises: Use the specified Venn diagrams to draw the following regions .
A∩B
B∪A
A0 ∩ B 0
A0 ∪ B
1
(A ∩ B) ∩ C
(A ∪ B) ∪ C
A0 ∩ (B 0 )0
(C 0 ∪ B) ∪ A0
(A ∪ B 0 ) ∪ C
A0 ∩ ∅
2
A∩U
U ∪A
(A ∩ B) ∪ B 0
∅0
Exercise: Draw Venn diagrams that represent the following situations.
(1) A, B, C ⊆ U , A ∩ B 6= ∅, B ∩ C 6= ∅, and A ∩ C = ∅.
(2) A, B, C ⊆ U , A ⊆ B, B ∩ C 6= ∅, and A ∩ C = ∅.
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(3) A, B, C ⊂ U , A ∩ B 6= ∅, A * B, B * A and C ⊆ A ∩ B.
(4) A, B, C ⊆ U , A ⊆ B and B ∩ C = ∅.
We can use Venn diagrams to help answer word problems. Try the following
problems
(1) In a class of students every student participates in the soccer team or the
debate team. 10 students only participate in the debate team, 31 students
only participate in the soccer team and 12 particiate in both teams.
Question: How many students are there in the class?
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(2) A survey of 60 people was taken and the following results were seen:
– 12 customers drank only tea and coffe
– 6 customers drank only juice
– 29 customers drank tea
– 2 customers drank only tea and juice
– 10 customers drank tea, coffee and juice
– 33 customers drank coffee
– 1 customer drank only juice and coffee
Question: How many customers drank juice?
Question: How many customers did not drink juice, tea or coffee?
Question: How many customers drank only coffee?
Question: How many customers drank only tea?
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Union Rule For Sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Union Rule For Disjoint Sets Assume that A and B are disjoint sets then
Exercise 38 (Chapter 7.2): Market research on cola consumption was conducted in a small town. The following results were seen
Age
21-25 (Y)
26-35 (M)
Over 35 (O)
Totals
Drink Regular
Cola (R)
40
30
10
80
Drink Diet
Cola (D)
15
30
50
95
Find the number of people in each set.
(1) Y ∩ R
(2) M ∪ (D ∩ Y )
(3) 00 ∪ N
(4) M ∩ D
(5) Y 0 ∩ (D ∪ N )
(6) M 0 ∩ (R0 ∩ N 0 )
(7) What does M ∪ (C ∩ Y ) mean in words ?
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Drink No
Cola (N)
15
20
10
45
Total
70
80
70
220
Exercise 57 (Chapter 7.2): A chicken farmer surveyed his flock with the
following results. The farm had
9 frat red roosters
15 red roosters
17 red hens
41 roosters
13
11
56
48
thin brown hens
thin red chickens (hens and roosters)
fat chickens
hens
Assume all chickens are either thin or fat, red or brown, and hens or roosters.
How many chickens were
(1) fat?
(2) fat roosters?
(3) thin and brown?
(4) red?
(5) fat hens?
(6) red and fat?
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