A Venn Diagram is a type of diagram that helps you organize groups of data that have some items in common.










Relationship between the set and its number of terms to calculate the union.
We can see that for 2 sets there are 3 terms needed to calculate the union, for 3 sets there are 7 terms, and for 4 sets there are 15 thems. How can we determine the relationship between the set and its number of terms to calculate the union?

Solution:

s = number of sets
i = stage

For example:
if s = 3:
# of Terms = 3C1 + 3C2 + 3C3
= 3 + 3 + 1
= 7
if s = 4:
# of Terms = 4C1 + 4C2 + 4C3 + 4C4
= 4 + 6 + 4 + 1
= 15
Therefore, we can determine the number of terms if there are 5 sets or more
if s = 5:
# of Terms = 5C1 + 5C2 + 5C3 + 5C4 + 5C5
= 5 + 10 + 10 + 5 + 1
= 31

Determining the formula to calculate the union of 5 set
From looking at the Princicples of Inclusion and Exclusion diagram above, we can see that the operator alternates from addition to subtraction between stages and the number of stages equal the number of sets. Following this pattern we see that:
5C1 = 5: first stage: n(a) + n(b) + n(c) + n(d) + n(e)
5C2 = 10: second stage: - n(a n b) - n(a n c) - n(a n d) - n(a n e) - n(b n c) - n(b n d) - n(b n e)
- n(c n d) - n(c n e) - n(d n e)
etc..

Following the above pattern, we can then determine that n(a U b U c U d U e) = n (a) + n(b) + n(c) + n(d) + n(e) - n(a n b) - n(a n c) - n(a n d) - n(a n e) - n(b n c) - n(b n d)
- n(b n e) - n(c n d) - n(c n e) - n(d n e) + n(a n b n c) + n(a n b n d) + n(a n b n e)
+ n(a n c n d) + n(a n c n e) + n(a n d n e) + n(b n c n d) + n(b n c n e) + n(b n d n e)
+ n(c n d n e) - n(a n b n c n d) - n(a n b n c n e) - n(a n b n d n e) - n(a n c n d n e)
- n(b n c n d n e) + n(a n b n c n d n e)


3 Set Example:

There are 900 employees at CantoCrafts Inc.Of these, 615 are female, 345 are under 35 years old, 482 are single, 295 are single females, 187 are singles under 35 years old, 190 are females under 35 years old, and 120 are single females under 35 years old. Use a Venn diagram to determine how many employees are married males who are at least 35 years old.

Solution:



n ( total number of singles, females or under 35 years of age) = n (females) + n (singles) + n (under 35 years of age) - n (females under 35) - n (single females) - n (singles uner 35) + n (single females under 35)
= 615 + 345 + 482 - 295 - 187 - 190 + 120
= 890

Total number of married men who are atleast 35 years old = Total number of employees - Total number of single females or employees under 35 years old
= 900 - 890
= 10

Therefore, 10 employees are married males uwho are atleast 35 years old.

4 Set Example:
Father Michael Goetz Student Services want to count the number of students who are taking Math, English, Religion or Science. They found that:

120 students are taking Science
120 students are taking English
130 students are taking Math
140 students are taking Religion
60 students are taking Science and English
60 students are taking English and Math
80 students are Math and Religion
60 students are taking Science and Religion
60 students are taking English and Religion
60 students are taking Science and Math
30 students are taking Science, English, Math
30 students are taking Science, Math, Religion
20 students are taking Science, English, Religion
40 students are taking English, Math, Religion
10 students are taking all four courses

Question: How many students are taking Science, English, Math or Religion?

Solution:

Let n(A) = Students taking Science
Let n(B) = Students taking English
Let n(C) = Students taking Math
Let n(D) = Students taking Religion

In order to solve this, the elements have to be grouped into 4 sets. Initially, one would think that a 4 circle Venn diagram would be able to display this information. However, it's impossible to visually display this information using four circles. To solve this, the four sets can be represeneted using a vertical rectangle, horizontal rectangle, a circle and 2 circles connected together as shown.





The sections in blue represents students taking only 1 of the 4 subjests
The sections in yellow represents students taking 2 of the 4 subjects
The sections in gray represents students taking 3 of the 4 subjects
The section in violet represents students taking all 4 subjects





n(A U B U C U D) = n(A) + n(B) + n(C) + n(D) - n(A n B) - n(B n C) - n(C n D) - n(A n D) - n(B n D) - n(A n C) + n(A n B n C) + n(A n C n D) + n(A n B n D) + n(B n C n D) - n(A n B n C n D)

= 120 + 120 + 130 + 140 - 60 - 60 - 80 - 60 - 60 - 60 + 30 + 30 + 20 + 40 - 10
= 240
Therefore, there are 240 Students taking Science, English, Math or Religion