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2
THE NATURE OF
SETS
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2.3
Applications of Sets
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Combined Operations with Sets
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Example 1 – Order of operations
Verbalize the correct order of operations and then illustrate
the combined set operations using Venn diagrams:
Solution:
a. This is a combined operation that should be read from
left to right. First find the complements of A and B and
then find the union. This is called a union of
complements.
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Example 1 – Solution
cont’d
Step 1 Shade A (vertical lines), then shade B (horizontal
lines).
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Example 1 – Solution
Step 2
cont’d
is every portion that is shaded with
horizontal or vertical lines. We show that here using
a color highlighter.
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Example 1 – Solution
cont’d
b. This is a combined operation that should be interpreted
to mean
which is the complement of the union.
First find A  B (vertical lines), and then find the
complement (color highlighter). This is called the
complement of a union. We show only the final result.
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De Morgan’s Laws
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De Morgan’s Laws
Notice from Example 1 that
If they were
equal, the final highlighted color portions of the Venn
diagrams would be the same. The next example takes us a
step further by showing what
does equal.
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Example 2 – De Morgan’s Law
Prove
Solution:
We use Pólya’s problem-solving guidelines for this
example.
Understand the Problem. We wish to prove the given
statement is true for all sets A and B, so we cannot work
with a particular example.
Devise a Plan. The procedure is to draw separate Venn
diagrams for the left and the right sides, and then to
compare them to see if they are identical.
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Example 2 – Solution
cont’d
Carry Out the Plan.
Step 1 Draw a diagram for the expression on the left side
of the equal sign, namely
. The final result is
shown with color highlighter.
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Example 2 – Solution
cont’d
Step 2 Draw a diagram for the expression on the right side
of the equal sign. First draw A (vertical lines) and
then B (horizontal lines). The final result,
is the part with both vertical and horizontal lines
(as
shown with the color highlighter) in the right portion
of Figure 2.13.
De Morgan’s Law
Figure 2.13
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Example 2 – Solution
cont’d
Step 3 Compare the portions shaded by the color
highlighter in the two Venn diagrams.
Look Back. They are the same, so we have proved
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De Morgan’s Laws
The result proved in Example 2 is called De Morgan’s law
for sets.
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Venn Diagrams with Three Sets
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Example 4 – Union is associative
If (P  Q)  R = P  (Q  R), we say that the operation of
union is associative. Is the operation of union for sets an
associative operation?
Solution:
We use Pólya’s problem-solving guidelines for this
example.
Understand the Problem. Even though we cannot answer
this question by using a particular example, we can use
one to help us understand the question.
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Example 4 – Solution
cont’d
Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},
P = {1, 4, 7},
Q = {2, 4, 9, 10}, and
R = {6, 7, 8, 9}. Is the following true?
(P  Q)  R = P  (Q  R)?
(P  Q)  R = {1, 2, 4, 7, 9, 10}  {6, 7, 8, 9}
= {1, 2, 4, 6, 7, 8, 9, 10}
(P  Q)  R = {1, 4, 7}  {2, 4, 6, 7, 8, 9, 10}
= {1, 2, 4, 6, 7, 8, 9, 10}
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Example 4 – Solution
cont’d
For this example, the operation of union for sets is
associative.
If we had observed (P  Q)  R  P  (Q  R), then we
would have had a counterexample. Although they are equal
in this example, we cannot say that the property is true for
all possibilities.
However, all is not lost because it did help us to understand
the question.
Devise a Plan. Use Venn diagrams.
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Example 4 – Solution
cont’d
Carry Out the Plan. Recall that the union is the entire
shaded area.
Look Back. The operation of union for sets is an
associative operation since the parts shaded in yellow are
the same for both diagrams.
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Survey Problems
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Survey Problems
There is a formula for the number of elements, but it is
easier to use Venn diagrams, as illustrated by Example 5.
The usual procedure is to fill in the number in the innermost
region first and work your way outward through the Venn
diagram using subtraction.
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Example 5 – Survey problem
A survey of 100 randomly selected students gave the
following information.
45 students are taking mathematics.
41 students are taking English.
40 students are taking history.
15 students are taking math and English.
18 students are taking math and history.
17 students are taking English and history.
7 students are taking all three.
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Example 5 – Survey problem
a. How many are taking only mathematics?
b. How many are taking only English?
c. How many are taking only history?
d. How many are not taking any of these courses?
Solution:
We use Pólya’s problem-solving guidelines for this
example.
Understand the Problem. We are considering students
who are members of one or more of three sets. If U
represents the universe, then | U| = 100.
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Example 5 – Solution
cont’d
We also define the three sets:
M = {students taking mathematics},
E = {students taking English},
H = {students taking history}.
Devise a Plan. The plan is to draw a Venn diagram, and
then to fill in the various regions. We fill in the innermost
region first, and then work our way outward (using
subtraction) until the numbers of elements of the eight
regions formed by the three sets are known.
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Example 5 – Solution
cont’d
Carry Out the Plan.
Step 1 We note |M  E  H| = 7 in Figure 2.16a.
First step is inner region
Figure 2.16a
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Example 5 – Solution
cont’d
Step 2 Fill in the other inner portions.
|E  H| = 17, but 7 have previously been accounted
for, so an additional 10 members (17 – 7 = 10) are
added to the Venn diagram (see Figure 2.16b).
|M  H| = 18; fill in 18 – 7 = 11;
|M  E| = 15; fill in 15 – 7 = 8.
Second step is two-region intersections
Figure 2.16b
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Example 5 – Solution
cont’d
Step 3 Fill in the other regions (see Figure 2.16c).
|H| = 40, but 28 have previously been accounted for
in the set H, so there are an additional 12 members
(40 – 11 – 7 – 10 = 12).
|E| = 41; fill in 41 – 8 – 7 – 10 = 16;
|M| = 45; fill in 45 – 11 – 7 – 8 = 19.
Third step is the one-region parts
Figure 2.16c
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Example 5 – Solution
cont’d
Step 4 Add all the numbers listed in the sets of the Venn
diagram to see that 83 students have been
accounted for. Since 100 students were surveyed,
we see that 17 are not taking any of the three
courses. (We also filled this number in Figure 16c.)
We now have the answers to the questions directly
from the Venn diagram:
a. 19 b. 16 c. 12 d. 17
Look Back. Does our answer make sense? Add all the
numbers in the Venn diagram as a check to see that we
have accounted for the 100 students.
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