1. CS 250, Discrete Structures, Fall 2013
Lecture 2.2: Set Theory
CS 250, Discrete Structures, Fall 2013
Nitesh Saxena
Adopted from previous lectures by Cinda Heeren

2. Course Admin HW1 Word Equation editor; Open Office Was just due
We will start to grade it
We will provide a solution set soon
Word Equation editor; Open Office
2/23/2019
Lecture Set Theory

3. Outline Set Theory, Operations and Laws 2/23/2019
Lecture Set Theory

4. Set Theory - Operators U A B like “exclusive or”
The symmetric difference, A  B, is: A  B = { x : (x  A  x  B) v (x  B  x  A)}
= (A - B) U (B - A) A U B
2/23/2019
Lecture Set Theory

5. Set Theory - Operators = (A - B) U (B - A) Proof:
A  B = { x : (x  A  x  B) v (x  B  x  A)}
= (A - B) U (B - A)
Proof:
{ x : (x  A  x  B) v (x  B  x  A)}
= { x : (x  A - B) v (x  B - A)}
= { x : x  ((A - B) U (B - A))}
= (A - B) U (B - A)
2/23/2019
Lecture Set Theory

6. Set Theory - Famous Laws
Two pages of (almost) obvious.
One page of HS algebra.
One page of new.
Don’t memorize them, understand them!
They’re in Rosen, p. 130
2/23/2019
Lecture Set Theory

7. Set Theory - Famous Laws
Identity
Domination
Idempotent
A  U = A
A U  = A
A U U = U
A   = 
A U A = A
A  A = A
2/23/2019
Lecture Set Theory

8. Set Theory - Famous Laws
Excluded Middle
Uniqueness
Double complement
A U A = U
A  A = 
A = A
2/23/2019
Lecture Set Theory

9. Set Theory – Famous Laws
Commutativity
Associativity
Distributivity
A U B =
B U A
B  A
A  B =
(A U B) U C =
A U (B U C)
A  (B  C)
(A  B)  C =
A U (B  C) =
A  (B U C) =
(A U B)  (A U C)
(A  B) U (A  C)
2/23/2019
Lecture Set Theory

10. Set Theory – Famous Laws
DeMorgan’s I
DeMorgan’s II
(A U B) = A  B
(A  B) = A U B
Venn Diagrams are good for intuition, but we aim for a more formal proof. p q
2/23/2019
Lecture Set Theory

11. 3 Ways to prove Laws or set equalities
Show that A  B and that A  B.
Use a membership table.
Use logical equivalences to prove equivalent set definitions.
New & important
Like truth tables
Not hard, a little tedious
2/23/2019
Lecture Set Theory

12. Example – the first way Prove that
() (x  A U B)  (x  A U B)  (x  A and x  B)  (x  A  B)
2. () (x  A  B)  (x  A and x  B)  (x  A U B)  (x  A U B)
(A U B) = A  B
2/23/2019
Lecture Set Theory

13. Example – the second way
Prove that using a membership table.
0 : x is not in the specified set
1 : otherwise
(A U B) = A  B A B
A  B
AUB
A U B 1
2/23/2019
Lecture Set Theory

14. Example – the third way
Prove that using logically equivalent set definitions.
(A U B) = A  B
(A U B) = {x : (x  A v x  B)}
= {x : (x  A)  (x  B)}
= {x : (x  A)  (x  B)}
= A  B
2/23/2019
Lecture Set Theory

15. Another example: applying the laws
X  (Y - Z) = (X  Y) - (X  Z). True or False?
Prove your response.
(X  Y) - (X  Z) = (X  Y)  (X  Z)’
= (X  Y)  (X’ U Z’)
= (X  Y  X’) U (X  Y  Z’)
=  U (X  Y  Z’)
= (X  Y  Z’)
= X  (Y - Z)
2/23/2019
Lecture Set Theory

16. A Proof (direct and indirect)
Pv that if (A - B) U (B - A) = (A U B) then
A  B = 
Suppose to the contrary, that A  B  , and that x  A  B.
Then x cannot be in A-B and x cannot be in B-A.
Then x is not in (A - B) U (B - A).
But x is in A U B since (A  B)  (A U B).
Thus, A  B = .
2/23/2019
Lecture Set Theory

17. Today’s Reading
Rosen 2.1 and 2.2
2/23/2019
Lecture Set Theory