Number Sets, Logical Operators, and
Venn Diagrams
Problem Solving
Defining Sets
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Sets {…}
Unions U
Intersections ∩
Compliment (uses the – symbol)
~ means “not”
Venn (Euler) diagrams
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Rectangle is the universe of numbers
Circles used for sets
Sets used in this example
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“Universe” of numbers {1,2,3,4}
Set A {1,2} and Set B {2,3}
Union
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Union, (A U B) = {1,2,3}
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Intersection
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Intersection, (A ∩ B) = {2}
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Not
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Complement, ~A = {3,4}
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Not Union
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Not Union, ~(A U B) = {4}
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Not Intersection
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Not Intersection, ~(A ∩ B) = {1,3,4}
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Yes, this is spelled
correctly; a “Compliment” is
something else.
Complement
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RecallUniverse={1,2,3,4}, A={1,2} and B={2,3}
Complement, (A – B) means everything in A except
the things that are also in B.
Venn Diagram
http://www.purplemath.com/modules/venndiag2.htm
Example Problem 1
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Universe={Anthony, Bruno, Charles, Diane,
Elaine, Fred, Gloria}
A={Anthony, Bruno, Charles, Diane}
B={Bruno, Elaine, Fred, Gloria}
Male={Anthony, Bruno, Charles, Fred}
Female={Diane, Elaine, Gloria}
Find A U Female
Find (Male – (A ∩ B))
Example Problem 2
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Using all integers from 1 to 10,
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–
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Set A = odd numbers
Set B = even numbers
Set C = numbers divisible by 3
Find B U C
Find A ∩ C
Find A – C
Example Problem 3
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There are 24 students in a class. 4 are in the
Civil program (C), 14 are in the Electrical
Power program (P), 6 are in Instrumentation
(I); 3 of which are double-majoring in
Instrumentation and Electrical Power (PI).
How many are not in Civil, Electrical Power,
nor Instrumentation?
Define the solution set
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Since we’re looking for students who do not
belong to the other sets, this is an instance to
use the “not” operator.
Your first reaction might be Civil + Power +
Instrumentation = 4 + 14 + 6, which seems to
account for all 24 students, but there are
three overlaps.
Venn Diagram
PPP
PPP
PPP
PP
CCCC
O
O
PI
PI
PI
O
Mathematically, ~(C U P U I)
I I
I
Applying Probability
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A fair coin is tossed three times and the
events A & B are defined as follows:
A: {at least one head is observed}
B: {the number of heads observed is odd}
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Identify the events in A, B, AUB, ~A, & A∩B
Find P(A), P(B), P(AUB), P(~A), and P(A∩B)
Are the events in A and B mutually
exclusive?
Identify the events for 3 coin tosses
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How many possible results are there?
What are they?
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Assign each result into the correct group:
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A:
B:
AUB:
~A:
A∩B
Draw a Venn Diagram
How likely is each set?
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Find probabilities:
P(A)
P(B)
P(AUB)
P(~A)
P(A∩B)
Are A and B mutually exclusive?
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If so, they would have no events in common