De Morgan's Laws
For any sets A and B,
(A ∩ B)' = A' U B'
AND
(A U B)' = A' ∩ B'
For example, if set A = {1,2,3,4,5,6}, and set B = {1,2,3,7,8,9}
L.S. = NOT (A intersects B) = NOT {1,2,3} = {4,5,6,7,8,9}
R.S. = (NOT A) union (NOT B) = {7,8,9} union {4,5,6} = {4,5,6,7,8,9}
L.S = R.S
Therefore, (A ∩ B)' = A' U B'
Cardinal Numbers and Surveys
Formula1: Cardinal Number Formula
For any two sets A and B,
n (A U B) = n (A) + n (B)  n (A ∩ B)
↑
Union of A and B counts the intersection area twice, so we subtract one of them
Surveys are word problems that apply various concepts of Set Theory.
Example1: The Board of Director of a university would like to know the number of students enrolled in math, arts and science courses. A survey of 144 firstyear students revealed the following facts:
58 students are taking at least one math course
63 students are taking at least one arts course
58 students are taking at least one science course
19 students are taking at least one math and one arts course
17 students are taking at least one math and one science course
4 students are taking at least one arts and one science course
1 student is taking math, arts and science courses
How many students are taking:
a) math course only?
b) science course only?
d) arts and science but not math course?
How many students are not taking any of the math, arts or science courses?
Solution1:
You should draw a Venn diagram to illustrate the various sets of data.
Set A be students who are taking math course(s)
Set B be students who are taking arts course(s)
Set C be students who are taking science course(s)
U be the universal set
A ∩ B ∩ C = 1 → # of students who are taking math, arts and science
A ∩ B ∩ C' = 191 = 18 → # of students who are taking math and arts but NOT science
A ∩ C ∩ B' = 171 = 16 → # of students who are taking math and science but NOT arts
B ∩ C ∩ A' = 41 = 3 → # of students who are taking arts and science but NOT math
A ∩ B' ∩ C' = 5818161 = 23 → # of students who are taking math course ONLY
B ∩ A' ∩ C' = 631831 = 41 → # of students who are taking arts course ONLY
C ∩ A' ∩ B' = 581631 = 38 → # of students who are taking science course ONLY
A U B U C = 1+18+16+3+23+41+38 = 140
Therefore, 4 students are not taking any of the math, arts or science courses.
