De Morgan’s Laws, Cardinal Numbers & Surveys

De Morgan’s Laws, Cardinal Numbers & Surveys

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 first-year 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?

c) arts and math but not science course?

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.

Let, 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' = 19-1 = 18 → # of students who are taking math and arts but NOT science

A ∩ C ∩ B' = 17-1 = 16 → # of students who are taking math and science but NOT arts

B ∩ C ∩ A' = 4-1 = 3 → # of students who are taking arts and science but NOT math

A ∩ B' ∩ C' = 58-18-16-1 = 23 → # of students who are taking math course ONLY

B ∩ A' ∩ C' = 63-18-3-1 = 41 → # of students who are taking arts course ONLY

C ∩ A' ∩ B' = 58-16-3-1 = 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.