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Chapter 3
Set Theory
Chapter 3 Set Theory
3.1 Sets and Subsets
A well-defined collection of objects
(the set of outstanding people, outstanding is very subjective)
finite sets, infinite sets, cardinality of a set, subset
A={1,3,5,7,9}
1  A,1  B ,1  C
B={x|x is odd}
C={1,3,5,7,9,...}
cardinality of A=5 (|A|=5)
A is a proper subset of B. A  B
CB
C is a subset of B.
Chapter 3 Set Theory
3.1 Sets and Subsets
Russell's Paradox
S  {A| A is a set and A  A}
( a) Show that is S  S , then S  S .
( b) Show that is S  S , then S  S
Principia Mathematica by Russel and Whitehead
Chapter 3 Set Theory
3.1 Sets and Subsets
subsets A  B  x[ x  A  x  B ]
A  B  x[ x  A  x  B ]
 x[ ( x  A)  x  B )]
 x[ x  A  x  B ]
set equality
C  D  ( C  D)  ( D  C )
C  D  (C  D  D  C )
 C  D D  C
Chapter 3 Set Theory
3.1 Sets and Subsets
null set or empty set : {}, 
universal set, universe: U
power set of A: the set of all subsets of A
A={1,2}, P(A)={, {1}, {2}, {1,2}}
If |A|=n, then |P(A)|=2n.
Chapter 3 Set Theory
3.1 Sets and Subsets
If |A|=n, then |P(A)|=2n.
For any finite set A with |A|=n0, there are C(n,k) subsets of
size k.
Counting the subsets of A according to the number, k, of
elements in a subset, we have the combinatorial identity
 n  n  n
 n
              2 n , for n  0
 0  1  2
 n
Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.9 Number of nonreturn-Manhattan paths between
two points with integer coordinated
From (2,1) to (7,4): 3 Ups, 5 Rights
R,U,R,R,U,R,R,U
8!/(5!3!)=56
permutation
8 steps, select 3 steps to be Up
{1,2,3,4,5,6,7,8}, a 3 element subset represents a way,
for example, {1,3,7} means steps 1, 3, and 7 are up.
the number of 3 element subsets=C(8,3)=8!/(5!3!)=56
Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.10 The number of compositions of an positive integer
4=3+1=1+3=2+2=2+1+1=1+2+1=1+1+2=1+1+1+1
4 has 8 compositions. (4 has 5 partitions.)
Now, we use the idea of subset to solve this problem.
Consider
4=1+1+1+1
The uses or not-uses of
1st plus 2nd plus 3rd plus
these signs determine
sign
sign
sign
compositions.
compositions=The number of subsets of {1,2,3}=8
Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.11 For integer n, r with n  r  1
prove  n  1   n   n 

   

 r   r   r  1
Let A  {x , a1 , a2 , , an }
combinatorially.
Consider all subsets of A that contain r elements.
 n  1  n  n 

    

 r   r   r  1
those include r
all possibilities
those exclude r
Chapter 3 Set Theory
3.1 Sets and Subsets
Ex. 3.13 The Pascal's Triangle
binomial
coefficients
 4
 
 0
 3
 
 0
 2
 
 0
 4
 
 1
 1
 
 0
 3
 
 1
 0
 
 0
 2
 
 1
 4
 
 2
 1
 
 1
 3
 
 2
 2
 
 2
 4
 
 3
 3
 
 3
 4
 
 4
Chapter 3 Set Theory
3.1 Sets and Subsets
common notations
(a) Z=the set of integers={0,1,-1,2,-1,3,-3,...}
(b) N=the set of nonnegative integers or natural numbers
(c) Z+=the set of positive integers
(d) Q=the set of rational numbers={a/b| a,b is integer, b not zero}
(e) Q+=the set of positive rational numbers
(f) Q*=the set of nonzero rational numbers
(g) R=the set of real numbers
(h) R+=the set of positive real numbers
(i) R*=the set of nonzero real numbers
(j) C=the set of complex numbers
Chapter 3 Set Theory
3.1 Sets and Subsets
common notations
(k) C*=the set of nonzero complex numbers
(l) For any n in Z+, Zn={0,1,2,3,...,n-1}
(m) For real numbers a,b with a<b,
[ a, b]  {x  R | a  x  b}
closed interval
(a, b)  {x  R | a  x  b}
open interval
[a, b)  {x  R | a  x  b}
(a, b]  {x  R | a  x  b}
half-open interval
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Def. 3.5 For A,B  U
union
a) A  B  {x | x  A  x  B}
intersection
b) A  B  {x | x  A  x  B}
c) AB  {x | x  A  B  x  A  B} symmetric difference
Def.3.6 mutually disjoint A  B  
A  U  A  {x | x U  x  A}
Def 3.7 complement
Def 3.8 relative complement of A in B
B  A  {x | x  B  x  A}
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Theorem 3.4 For any universe U and any set A,B in U, the
following statements are equivalent:
a) A  B
b) A  B  B
reasoning process
c) A  B  A
(a)  (b), (b)  (c),
d) B  A
(c)  (d), and (d)  (a)
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
The Laws of Set Theory
(1) A  A
Law of Double Complement
(2) A  B  A  B
Demorgan ' s Laws
A B  A B
(3) A  B  B  A
Commutative Laws
A B  B  A
(4) A  ( B  C )  ( A  B )  C Associative Laws
A  ( B  C )  ( A  B)  C
(5) A  ( B  C )  ( A  B )  ( A  C ) Distributi ve Laws
A  ( B  C )  ( A  B)  ( A  C )
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
The Laws of Set Theory
(6) A  A  A, A  A  A Idempotent Laws
(7) A    A, A  U  A Identity Laws
(8) A  A  U , A  A  
(9) A  U  U , A   = 
Inverse Laws
Domination Laws
(10) A  ( A  B )  A
A( A B )  A
Absorption Laws
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
s
dual of s (sd)

U
U





Theorem 3.5 (The Principle of Duality) Let s denote a theorem
dealing with the equality of two set expressions. Then sd is also
a theorem.
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Ex. 3.17 What is the dual of
A B?
Since A  B  A  B  B . The dual of A  B is the dual of
A  B  B , which is A  B  B . That is, B  A.
A B
Venn diagram
U
A
A
A
B
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Ex. 3.19. Negate A  B .
A  B  {x | x  A  x  B}  A  B
 A  B  A B  A B
Ex. 3.20 Negate AB .
 AB  {x | x  A  B  x  A  B}
 ( A  B)  ( A  B)  ( A  B)  ( A  B)
 AB  ( A  B )  ( A  B )  A  B  ( A  B )
 ( A  B )  ( A  B )  [( A  B )  A]  [( A  B )  B ]
 ( B  A)  ( A  B )  ( A  B )  ( A  B )
 AB  A B
Chapter 3 Set Theory
3.2 Set Operations and the Laws of Set Theory
Def 3.10.  Ai  {x | x  Ai for at least one i  I }, and
iI
 Ai  {x | x  Ai for every i  I }
iI
I: index set
Theorem 3.6 Generalized DeMorgan's Laws
 Ai   Ai
iI
iI
 Ai   Ai
iI
iI
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
Ex. 3.23. In a class of 50 college freshmen, 30 are studying
BASIC, 25 studying PASCAL, and 10 are studying both. How
many freshmen are studying either computer language?
U
5
A
B
| A  B |  | A| | B | | A  B |
20
10
15
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
Ex 3.24. Defect types of an AND gate:
D1: first input stuck at 0
11
D2: second input stuck at 0
D3: output stuck at 1
A
4
5
12
3 7
15
B
43
C
Given 100 samples
set A: with D1
| A  B  C |  | A| | B | | C | | A  B |
set B: with D2
 | A  C | | B  C | | A  B  C |
set C: with D3
with |A|=23, |B|=26, |C|=30,
| A  B |  7, | A  C |  8, | B  C |  10,
| A  B  C |  3 , how many samples have defects? Ans:57
Chapter 3 Set Theory
3.3 Counting and Venn Diagrams
Ex 3.25
There are 3 games. In how many ways can one play
one game each day so that one can play each of the
three at least once during 5 days?
set A: without playing game 1
set B: without playing game 2
balls
containers
set C: without playing game 3
1
| A|  | B |  | C |  2 5
g1
2
5
g2
| A  B |  | B  C |  | C  A|  1
3
| A B  C| 0
g3
4
5
5
| A  B  C |  3  2  3  1  0  93
5
5
Ans  3  93  150
Chapter 3 Set Theory
3.4 A Word on Probability
event A
a
elementary event
U=sample space
Pr(A)=the probability that A occurs=|A|/|U|
Pr(a)=|{a}|/|U|=1/|U|
Chapter 3 Set Theory
3.4 A Word on Probability
Ex. 3.27 If one tosses a coin four times, what is the probability
of getting two heads and two tails?
Ans: sample space size=24=16
event: H,H,T,T in any order, 4!/(2!2!)=6
Consequently, Pr(A)=6/16=3/8
Each toss is independent of the outcome of any previous toss.
Such an occurrence is called a Bernoulli trial.