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Section 2.4 – Set Operations and Venn Diagrams with Three Sets
Objective #1: Perform operations with three sets.
In this section, we will perform the operations union, intersection, and
finding the complement of sets to a situations involving three sets. It will be
important to do the operations in the correct order by performing any set
operations inside parentheses first.
Given the sets below, find the following:
Ex. 1 Given: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
A = {2, 3, 5, 7, 11}
B = {1, 3, 5, 7, 9}
C = {1, 2, 3, 5, 8}
Find:
a) C ∪ (A ∩ B)
b) (C ∪ A) ∩ (C ∪ B)
c) A ∩ (B' ∪ C')
Solution:
a) First, find A ∩ B:
A ∩ B = {2, 3, 5, 7, 11} ∩ {1, 3, 5, 7, 9} = {3, 5, 7}
Now, find C ∪ (A ∩ B):
C ∪ (A ∩ B) = {1, 2, 3, 5, 8} ∪ {3, 5, 7} = {1, 2, 3, 5, 7, 8}
b) First, find (C ∪ A):
(C ∪ A) = {1, 2, 3, 5, 8} ∪ {2, 3, 5, 7, 11} = {1, 2, 3, 5, 7, 8, 11}
Next, find (C ∪ B):
(C ∪ B) = {1, 2, 3, 5, 8} ∪ {1, 3, 5, 7, 9} = {1, 2, 3, 5, 7, 8, 9}
Finally, find (C ∪ A) ∩ (C ∪ B):
(C ∪ A) ∩ (C ∪ B) = {1, 2, 3, 5, 7, 8, 11} ∩ {1, 2, 3, 5, 7, 8, 9}
= {1, 2, 3, 5, 7, 8}
c) To find B', take the set U and cross out the elements from B:
B' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 2, 4, 6, 8, 10, 11}
To find C', take the set U and cross out the elements from C:
C' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 4, 6, 7, 9, 10, 11}
Next, find B' ∪ C':
B' ∪ C' = {0, 2, 4, 6, 8, 10, 11} ∪ {0, 4, 6, 7, 9, 10, 11}
= {0, 2, 4, 6, 7, 8, 9, 10, 11}
Finally, find A ∩ (B' ∪ C'):
A ∩ (B' ∪ C') = {2, 3, 5, 7, 11} ∩ {0, 2, 4, 6, 7, 8, 9, 10, 11}
= {2, 7, 11}
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Objective #2: Venn Diagrams with three sets.
With three sets, the Venn Diagrams are more complicated. The three sets
may have elements in common, two of the three sets may have elements in
common, each set may have elements that are not elements of the other
two sets, and there may be elements that are in none of the sets. The three
sets will therefore divide the Venn Diagram into eight different regions.
U
A
I
B
II
III
V
IV
VI
VII
C
Region
I
II
III
IV
V
VI
VII
VIII
VIII
Elements
The elements that are in A only: A ∩ (B ∪ C)'
The elements that are in A and B, but not C: A ∩ B ∩ C'
The elements that are in B only: B ∩ (A ∪ C)'
The elements that are in A and C, but not B: A ∩ C ∩ B'
The elements that are in A, B, and C: A ∩ B ∩ C
The elements that are in B and C, but not A: B ∩ C ∩ A'
The elements that are in C only: C ∩ (A ∪ B)'
The elements that are not in any of the sets: (A ∪ B ∪ C)'
Constructing a Venn Diagram:
1)
Start in the middle where all three sets overlap (Region V).
2)
Next, complete the sections where two of the three, but not all three
sets overlap (Regions II, IV, & VI).
3)
Now, complete the sections where one set does not have any
elements in common with the other two sets (Regions I, III, & VII).
4)
Finally, complete the section with the elements in none of the sets
(Region VIII).
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Use set operations to construct the appropriate Venn Diagram:
Ex. 2 Given: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
A = {2, 3, 5, 7, 11}
B = {1, 3, 5, 7, 9}
C = {1, 2, 3, 5, 8}
Solution:
1) Start with Region V: A ∩ B ∩ C
A ∩ B ∩ C = {2, 3, 5, 7, 11} ∩ {1, 3, 5, 7, 9} ∩ {1, 2, 3, 5, 8} = {3, 5}
Write 3 and 5 in Region V.
U
A
B
I
IV
II
3
III
5
VII
C
VI
VIII
2) Next, find Regions II, IV, & VI:
Region II: A ∩ B ∩ C'
A ∩ B = {2, 3, 5, 7, 11} ∩ {1, 3, 5, 7, 9} = {3, 5, 7}
C' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 4, 6, 7, 9, 10, 11}
A ∩ B ∩ C' = {3, 5, 7} ∩ {0, 4, 6, 7, 9, 10, 11} = {7}
Write 7 in Region II.
Region IV: A ∩ C ∩ B'
A ∩ C = {2, 3, 5, 7, 11} ∩ {1, 2, 3, 5, 8} = {2, 3, 5}
B' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 2, 4, 6, 8, 10, 11}
A ∩ C ∩ B' = {2, 3, 5} ∩ {0, 2, 4, 6, 8, 10, 11} = {2}
Write 2 in Region IV.
Region VI: B ∩ C ∩ A'
B ∩ C = {1, 3, 5, 7, 9} ∩ {1, 2, 3, 5, 8} = {1, 3, 5}
A' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 1, 4, 6, 8, 9, 10}
B ∩ C ∩ A' = {1, 3, 5} ∩ {0, 1, 4, 6, 8, 9, 10} = {1}
Write 1 in Region VI.
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U
A
B
I
III
7
3
5
2
1
VII
C
VIII
3) Now, find Regions I, III, & VII:
Region I: A ∩ (B ∪ C)'
B ∪ C = {1, 3, 5, 7, 9} ∪ {1, 2, 3, 5, 8} = {1, 2, 3, 5, 7, 8, 9}
(B ∪ C)' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 4, 6, 10, 11}
A ∩ (B ∪ C)' = {2, 3, 5, 7, 11} ∩ {0, 4, 6, 10, 11} = {11}
Region III: B ∩ (A ∪ C)'
(A ∪ C) = {2, 3, 5, 7, 11} ∪ {1, 2, 3, 5, 8} = {1, 2, 3, 5, 7, 8,11}
(A ∪ C)' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 4, 6, 9, 10}
B ∩ (A ∪ C)' = {1, 3, 5, 7, 9} ∩ {0, 4, 6, 9, 10} = {9}
Region VII: C ∩ (A ∪ B)'
A ∪ B = {2, 3, 5, 7, 11} ∪ {1, 3, 5, 7, 9} = {1, 2, 3, 5, 7, 9, 11}
(A ∪ B)' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} = {0, 4, 6, 8, 10}
C ∩ (A ∪ B)' = {1, 2, 3, 5, 8} ∩ {0, 4, 6, 8, 10} = {8}
Write 11 in Region I, 9 in Region III, and 8 in Region VII.
U
A
B
11
7
3
9
5
2
1
8
C
VIII
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4) Finally, find Region VIII: (A ∪ B ∪ C)'
A ∪ B ∪ C = {2, 3, 5, 7, 11} ∪ {1, 3, 5, 7, 9} ∪ {1, 2, 3, 5, 8}
= {1, 2, 3, 5, 7, 8, 9, 11}
(A ∪ B ∪ C)' = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
= {0, 4, 6, 10}
Write 0, 4, 6, and 10 in Region VIII.
U
A
B
11
7
3
9
5
2
1
8
C
0
4
6
10
Construct the appropriate Venn Diagram:
Ex. 3 Given: U = {a, b, d, e, f, k, m, n, p, x, y}
A = {a, b, d, e, f}
B = {a, d, k, n, p}
C = {a, e, f, p, x}
Solution:
Let's do this without using set operations.
1) Since a is the only element in common in all three sets, then its
goes in region V.
2) a and d are the only elements in common with both A and B, but a
is already in region V, so d has to go in region II.
a, e, and f are the only elements in common with both A and C,
but a is already in region V, so e and f have to go in region IV.
a and p are the only elements in common with both B and C, but a
is already in region V, so p has to go in region VI.
Let's take a look at our Venn Diagram:
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U
A
B
d
a
e f
p
C
3)
Elements a, b, d, e, & f are in A, but a, d, e, & f are already in
other regions of A, so b has to go in Region I.
Elements a, d, k, n, & p are in B, but a, d, & p are already in
other regions of B, so k & n have to go in Region III.
Elements a, e, f, p, & x are in C, but a, e, f, & p are already in
other regions of C, so x has to go in Region VII.
U
A
b
B
d
k n
a
e f
p
x
C
4)
Finally, the elements a, b, d, e, f, k, m, n, p, x, & y are in the
universal set, a, b, d, e, f, k, n, p, & x are already on the
diagram, so m and y have to go in Region VIII.
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U
A
b
B
d
k n
a
e f
p
x
C
m
y
Use the Venn Diagram below to find the following sets:
Ex. 4
U
A
x p
B
r s
w t
b f
e g
m n
h k
C
q
a) A
b) A ∩ C
c) (B ∪ C)'
v
d) (A ∪ B) ∩ C'
Solution:
a) The set A is all the elements in Regions I, II, IV, and V:
A = {b, e, f, g, p, r, s, x}
b) The set A ∩ C is where sets A and C overlap which occurs in
Regions IV and V:
A ∩ C = {b, e, f, g}
c) The set B ∪ C is all the elements in Regions II, III, IV, V, VI,
and VII. So, (B ∪ C)' are the Regions I and VIII:
(B ∪ C)' = {p, q, v, x}
d) The set (A ∪ B) ∩ C' is all the elements in the union of A and B,
but outside of C which are the Regions I, II, and III:
(A ∪ B) ∩ C' = {p, r, s, t, w, x}
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Objective #3: Using Venn Diagrams to Prove the Equality of Sets.
We can also use Venn Diagrams to prove that two sets are equal by
showing that each set are represented by the same regions on the Venn
Diagram. In doing so, we are using Deductive Reasoning.
Use Venn Diagrams to prove the following are equal:
Ex. 5
(A' ∩ B)' = A ∪ B'
Solution:
First, draw a Venn diagram:
U
A
I
B
II
III
IV
Next, identify the regions that represent (A' ∩ B)'
Set
Regions in the Venn Diagram
A'
III & IV (regions outside of A)
B
II & III (regions of B)
A' ∩ B III (region that A' and B have in common)
(A' ∩ B)' I, II, & IV (regions outside of A' ∩ B)
Now, identify the regions that represent A ∪ B'
Set
Regions in the Venn Diagram
A
I & II (regions of A)
B'
I & IV (regions outside of B)
A ∪ B' I, II, & IV (regions of A and B' put together)
Both (A' ∩ B)' and A ∪ B' are represented by the same regions on the
Venn Diagram, so (A' ∩ B)' = A ∪ B' for all sets in the universal set
U.
Ex. 6
B ∩ (A ∪ C) = (A ∩ B) ∪ (B ∩ C)
Solution:
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First, draw a Venn diagram:
U
A
I
B
II
III
V
IV
VI
VII
C
VIII
Next, identify the regions that represent B ∩ (A ∪ C):
Set
Regions in the Venn Diagram
A
I, II, IV, & V (regions of A)
C
IV, V, VI, & VII (regions of C)
A∪C
I, II, IV, V, VI, & VII (region that A and C put together)
B
II, III, V, & VI (regions of B)
B ∩ (A ∪ C)
II, V, & VI (regions that B and A ∪ C have in common)
Now, identify the regions that represent (A ∩ B) ∪ (B ∩ C):
Set
Regions in the Venn Diagram
A
I, II, IV, & V (regions of A)
B
II, III, V, & VI (regions of B)
A∩B
II & V (regions that A and B put have in common)
B
II, III, V, & VI (regions of B)
C
IV, V, VI, & VII (regions of C)
B∩C
V, & VI regions that B and C put have in common)
(A ∩ B) ∪ (B ∩ C) II, V, & VI (regions that A ∩ B and B ∩ C have in
common)
Both B ∩ (A ∪ C) and (A ∩ B) ∪ (B ∩ C) are represented by the
same regions on the Venn Diagram, so
B ∩ (A ∪ C) = (A ∩ B) ∪ (B ∩ C) for all sets in the universal set U.
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Objective #4:
Write sets represented by Venn Diagrams.
Use the symbols A, B, C, ∩, ∪, and ' to describe each shaded region:
Ex. 7
Solution:
The shaded region represents the elements in A or B, but not in both.
So, we need to take the elements in A ∪ B and remove the elements
in A ∩ B. We can do this by intersecting the A ∪ B with the
complement of A ∩ B:
(A ∪ B ) ∩ (A ∩ B)'
Ex. 8
Solution:
The shaded region represents the elements in A or in the intersection
of B and C. So, we can take the union of A with the intersection of B
and C to get the desired result:
A ∪ (B ∩ C)