1. Discrete Mathematics Dr.-Ing. Erwin Sitompul
Lecture 4
2. SETS
Discrete Mathematics
Dr.-Ing. Erwin Sitompul

2. Homework 3
Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 } as a set universe and the sets :
A = { 1, 2, 3, 4, 5 },
B = { 4, 5, 6, 7 },
C = { 5, 6, 7, 8, 9 },
D = { 1, 3, 5, 7, 9 },
E = { 2, 4, 6, 8 },
F = { 1, 5, 9 }.
Determine:
a) A  C
b) A  B
c) A  F
d) (C  D)  E
e) (F – C) – A

3. Solution of Homework 3 U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 },
A = { 1, 2, 3, 4, 5 },
B = { 4, 5, 6, 7 },
C = { 5, 6, 7, 8, 9 },
D = { 1, 3, 5, 7, 9 },
E = { 2, 4, 6, 8 },
F = { 1, 5, 9 }.
a) A  C
b) A  B
c) A  F
d) (C  D)  E
e) (F – C) – A
= { 1,2, 3, 4, 5, 6, 7, 8, 9 } = U
= { 4, 5 }
= { 6, 7, 8, 9 }  { 1, 5, 9 }
= { 9 }
= { 1, 3, 6, 8 }  { 2, 4, 6, 8 }
= { 1, 2, 3, 4 }
= { 1 } – { 1, 2, 3, 4, 5 }
= 

4. Principle of Duality Example:
In USA, the steering wheel of a car is located on the left hand side of the front row. In England, it is on the right.
Traffic rules: In USA:
People drive the car on the right half of the road.
The left lane is for overtaking only.
In crossroad, if you want to turn right, you do not have to wait for the green light.
In England:
People drive the car on the left half of the road.
The right lane is for overtaking only.
At crossroad, if you want to turn left, you do not have to wait for the green light.
In this case, the Principle of Duality applies:
The concept of left and right for the traffic rules in one country can be exchanged to obtain the rule for the other country.

5. Principle of Duality
Duality Principle applies when two different concepts are interchangeable while still delivering the true answer.
Suppose E is an equation of set algebra. Then the dual E* of E is the equation obtained by replacing each occurrence of , , U and  in E by , , , and U, respectively.
For example, the dual of
(U  A)  (B  A) = A is (  A)  (B  A) = A

6. Duality in Set Algebra
  
  
  U
U  
Dual
Dual
Dual
Dual

7. Duality in Set Algebra         U Dual U   Dual Dual Dual

8. Duality in Set Algebra Example: The dual of (A  B)  (A  B) = A is:

9. Inclusion-Exclusion Principle
For any sets A and B, the following set equation apply:
A  B = A + B – A  B
A  B = A + B – 2A  B
A  B
A  B

10. Inclusion-Exclusion Principle
Example:
In a poll among 40 students, it comes out that 32 of them prefer Internet Explorer, 18 students like Mozilla Firefox more, and 2 students do not like either of the software. Determine:
a) The number of students who like Internet Explorer or Mozilla Firefox.
b) The number of students who like Internet Explorer or Mozilla Firefox, but not both of them.
Solution:
Suppose
U = { The number of students participating in the poll}
A = { The number of students who like Internet Explorer }
B = { The number of students who like Mozilla Firefox }
Then
U= 40, A= 32, B= 18, A  B= 2

11. Inclusion-Exclusion Principle
a) The number of students who like Internet Explorer or Mozilla Firefox.
A  B = U – A  B = 40 – 2 = 38
b) The number of students who like Internet Explorer or Mozilla Firefox, but not both of them.
A  B = A + B – A  B = – 38 = 12
A  B = A + B – 2A  B= – 212 = 26
AB
AB

12. Inclusion-Exclusion Principle
For any sets A, B, and C the following set equation apply:
A  B  C = A + B + C – A  B – A  C –
B  C + A  B  C

13. Inclusion-Exclusion Principle
Example:
Among integers between (and including) 101 and 600, how many numbers are not divisible by 4 and 5 or divisible by 4 and 5?
Solution:
Suppose
U = { The number of integers between (and including) and 600 }
A = { Members of U, divisible by 4 }
B = { Members of U, divisible by 5 }
Then
U= 500
A= 500/4 = 125
B= 500/5 = 100
A  B = 500/20 = 25
Question: A  B?
p  q  (p  q)  ~(p  q)
~(p  q)  (~p  ~q)  (p  q)

14. Inclusion-Exclusion Principle
A= 500/4 = 125
B= 500/5 = 100
A  B = 500/20 = 25
A  B = A + B – 2A  B
= – 225
= 175
A  B = U – A  B
= 500 – 175
= 325

15. Proving of Set Proposition
Set Proposition is an argument built by using set notation.
Generally, propositions are written in equality and are subject to prove.
Example:
Prove that “A  (B  C) = (A  B)  (A  C).”
Proving set proposition can be conducted in 2 ways:
1. By using membership table.
2. By using laws of set algebra.

16. Proving by Using Membership Table
Example:
Prove that “A  (B  C) = (A  B)  (A  C).”
1 : Not a null set (T)
0 : Null set (F)
Proven that A  (B  C) = (A  B)  (A  C)

17. Proving by Using Laws of Set Algebra
Example:
Prove that “(A  B)  (A  B) = A.”
Solution:
(A  B)  (A  B) = A  (B  B) Distributive Laws
= A  U Complement Laws
 = A Identity Laws
Example:
Prove that “A  (B – A) = (A  B).”
Solution:
A  (B – A) = A  (B  A) Definition of Difference
= (A  B)  (A  A) Distributive Laws
= (A  B)  U Complement Laws
 = A  B Identity Laws

18. Homework 4
Prove that for arbitrary sets A and B, the following set equation apply:
a) A  (A  B) = A  B
b) A  (A  B) = A  B

19. Homework 4
New
Prove that for arbitrary sets A, B, and C, the following set equation apply:
a) A  (B  C) = (C  B)  A
b) (B – A)  (C – A) = (B  C) –A