More with Sets and Venn
Diagrams
Problem Solving & Critical Thinking
Venn Diagrams
• Using Sets (may be numbers, letters, words,
etc.)
Set A
Set B
Universe
• Any of the elements that are considered, and
don’t necessarily have to be within any set
Union
• Anything that is contained by any of the
defined sets
• A B for two sets
Intersection
• Any elements common to all sets being
considered
• A B
Complement
• All elements not within the set
A’
B’
Subtraction Operators
• Start with a set, and then remove others
• A-B
Subtraction Operators
• B–A
• {2,5,6,7,9} – {1,3,5,7,9} = {2,6}
Cardinality
• The number of elements within the set
• Denoted as A
• A = 5 in this example
8
5
3
1
9
2
7
6
4
Cardinality
• What is A’ ?
• What is A-B ?
Example Problem
• A = Students who hunt with a bow
• B = Students who hunt with a gun
• Assume there are 50 students who hunt with
a gun, 30 who hunt with a bow, and 65 who
hunt with at least one of them
• How many students do both?
Givens:
• A = 30
• B = 50
• A B = 65
Want to Solve
• How many students like both?
• A B=A+B-A B
=
+
• So A B = A + B - A B
• 50 + 30 – 65 = 15
-
You Try
• In a random sample of MS patients, it was
found that 25 had allergies to milk products,
32 had allergies to peanuts, and 15 had
allergies to both. How many had allergies to
at least one of those two things?
Three Sets
•
•
•
•
•
•
•
•
A = Students in Spanish class; A =125
B = People who own Halo 4; B = 40
C = Students who have their own apartment; C =140
2 students in Spanish class have Halo 4
88 students in Spanish class have their own apartment
16 students with Halo 4 have their own apartment
200 students fall into at least one of these categories
Problems:
– Describe students in A
– Find A B C
B
C
Sketch Out the Venn Diagram
Mathematically
• A B C
• Number of times each area is counted
1
1
1
1
1
1
1
Mathematically
• A+B+C
+
• Number of times each area is counted
1
1
2
2
3
1
2
Mathematically
• A B+A C+B C
+
• Number of times each area is counted
0
0
1
1
3
0
1
Mathematically
• A B C
• Number of times each area is counted
0
0
0
0
1
0
0
Mathematically
• A B C = A+B+C - (A B+A C+B C) + A B C
=
A B C
-
A+B+C
+
A B+A C+B C
A B C
• A B C=A+B+C-A B-A C-B C+A B C
• Solve for A B C :
• A B C-A-B-C+A B+A C+B C=A B C
Variables in the Equation
•
•
•
•
•
•
•
A
B
C
A
A
B
A
= 125
= 40
= 140
B =2
C = 88
C = 16
B C = 200
}
Hint: A B C cannot be larger than any of
these intersections
Write out Equation and Solve
• A B C-A-B-C+A B+A C+B C=A B C
•
200 - 125 – 40 - 140+ 2 + 88 + 16 = 1
1
Venn Diagram
• Using the Intersections first:
1
87
• Then applying the original sets:
36
87
1
1
1
37
• Check: 36+1+1+23+87+15+37=200
• A B C =200 (which was given)
15
23
15