Download The Principle of Inclusion and Exclusion

MDM4U1
1.5 Counting with Venn Diagrams
Combinations
SETS AND SUBSETS
A set is a collection of distinct objects. The objects in the set are called elements or members of the set.
The members of a set are listed in the following manner:
A  {a , e,i ,o ,u }
B  {Ford , Chrysler , Nissan ,Toyota } N  {1,2,3, 4,....}
Often we are interested in the number of elements in a set rather than the individual items.
To denote the number of members of a set, we use the notation n(A).
DEFINITIONS:
Disjoint sets: Two or more sets that have no elements in common.
Subset: If all of the elements of set B are contained in set A, then B is said to be a subset of A.
B A
Intersection of Sets: The intersection of two sets is the set of elements which are common to both sets.
A and B
or
A
B
Union of Sets: The union of two sets A and B is the set of all elements that are in A, B or both.
A or B
or
A
B
Example 1:
A  {1,2,3, 4,5,...20}, B  {2, 4,6,8} C  {5,10,15,20,25} D  {1, 4,9,16,25,36}
(a) Find n A  
n C  
n B  
n D  
(b) Name any disjoint sets.
(c) Are there any subsets?
(d) Find each of the following:
A and C 
A
B or C 
C
D
D 
B
C 
A
B
C
MDM4U1
1.5 Counting with Venn Diagrams
Combinations
Just as it is possible to graph relationships between variables such as y  2x 2  1 , we can illustrate the
relationship between sets with a Venn Diagram.
Example 2:
Disjoint Sets:
The sets A and B have no elements in common and are both
subsets of a universal set that is denoted by S:
Intersecting Sets:
A and B have some elements in common which are in the
overlapping area of the two circles. Shade the region A B of
the Venn Diagram.
S
A
B
S
Union of Sets:
The union of A and B is all the elements in A or B. Shade the
region A B in the Venn Diagram.
S
Subsets:
B  A means B is a subset of A. Every element in B is also in A.
S
A
B
A
B
A
Intersection of Three Sets:
The intersection of the three sets would be the area represented
by
B
S
B
A
The union of the three sets would be the area represented by
C
MDM4U1
1.5 Counting with Venn Diagrams
Combinations
The Principle of Inclusion and Exclusion
Example 1:
There are 12 students on the volleyball team (set A) and 15 on the basketball team (set B). There are 5
students who play on both teams. Draw a Venn diagram to illustrate these sets.
Explain what each of the following means and give a numerical answer.
a) n A and B 
b) n A or B 
In general: For sets A and B, the number of elements in either A or B is:
Example 2:
There are 20 students in this class. If 15 take grade 12 English, 10 take History and 3 take neither, how
many take both Grade 12 English and History?
MDM4U1
1.5 Counting with Venn Diagrams
Combinations
The Principle of Inclusion and Exclusion with 3 Sets
Example 3:
According to a representative of Student Services, of 100 students currently taking Grade 12 courses, the
distribution of students in Mathematics courses is as follows: Calculus and Vectos 52, Functions 38, Data
Management 41, Functions and Data Management 17, Functions and Calculus 22, Data Management and
Calculus 15, and there are 8 students taking all 3 courses.
(a)
Draw a Venn diagram to illustrate this problem.
S
(b)
How many study only Functions?
(c)
How many students are taking Math courses at the Grade 12 level?
Principle of Inclusion and Exclusion: