Chapter 5, Set Theory
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Set definitions
Definition of a set:
name of set = {list of elements, or a description of the elements}
Examples: A = {1,2,3} or B = {x  Z | 4 < x < 4} or
C = {x  Z+ | 4 < x < 4}
A set is completely defined by its elements, i.e.,
{a,b} = {b,a} = {a,b,a} = {a,a,a,b,b,b}
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Discrete Structures
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Lecture 28
April 7, 2008
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More set concepts
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The universal set (U) is the set consisting of all
possible elements in some particular situation under
consideration
A set can be finite or can be infinite
For a set S, n(S) or |S| are used to refer to the
cardinality of S, which is the number of elements in S
The symbol  means "is an element of"
The symbol  means "is not an element of"
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Subset
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A  B  (x  U)[x  A  x  B]
A is contained in B
B contains A
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A  B  (x  U)[x  A  x  B]
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Relationship between membership and subset:
(x  U)[x  A  {x}  A]
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Definition of set equality: A = B  A  B  B  A
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Do these represent the same sets or not?
X = {x  Z | (p  Z)[x = 2p]}
Y = {y  Z | (q  Z)[y = 2q  2]}
A = {x  Z | (i  Z)[x = 2i + 1]}
B = {x  Z | (i  Z)[x = 3i + 1]}
C = {x  Z | (i  Z)[x = 4i + 1]}
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Formal definitions of set operations
Union:
A  B  {x U | x  A  x  B}
Intersection:
A  B  {x U | x  A  x  B}
Complement:
A  A'  A  {x  U | x  A}
Difference:
A  B  {x U | x  A  x  B}
c
A  B  A  B'
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Venn diagrams
Sets are represented as regions (usually circles) in the
plane in order to graphically illustrate relationships
between them.
A
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B
A
B
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The empty set and its properties
The empty set  has no elements, so  = {}.
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( sets X)[  X]
There is only one empty set.
( sets X)[X   = X]
( sets X)[X  X' = ]
( sets X)[X   = ]
U' = 
' = U
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Discrete Structures
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Lecture 29
April 9, 2008
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Ordered n-tuples
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An ordered n-tuple takes order and multiplicity into
account
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The tuple (x1,x2,x3,…,xn)
– has n values
– which are not necessarily distinct
– and which appear in the order listed
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(x1,x2,x3,…,xn) = (y1,y2,y3,…,yn)  (i 1  i  n)[xi = yi ]
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2-tuples are called pairs, and 3-tuples are called triples
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The Cartesian product
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The Cartesian product of sets A and B is defined as
A  B  {(a, b) | a  A  b  B}
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n(A  B) = n(A)  n(B)
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Proper subset
A B  A B A B
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Disjoint sets
A and B are disjoint
 A and B have no elements in common
 (x  U)[x  A  x  B  x  B  x  A]
A  B =   A and B are disjoint sets
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Power set
P (A) = the set of all subsets of A
Examples- what are P ({a})?
P ({a,b,c})?
P ()?
P ({})?
P ({,{}})?
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Properties of sets in Theorem 5.2.1
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Inclusion
A B  A
A  A B
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Transitivity
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A B  B
B  A B
A BBC  AC
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Discrete Structures
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Lecture 30
April 11, 2008
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Properties of sets in Theorem 5.2.2
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DeMorgan’s for complement
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Distribution of union and intersection
( A  B)'  A' B'
( A  B)'  A' B'
A  ( B  C )  ( A  B)  ( A  C )
A  ( B  C )  ( A  B)  ( A  C )
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There are a number of others as well; see the text or the
handout of logical rules and equivalences for the full list
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Using Venn diagrams to help find
counterexamples
A  ( B  C )  ?  ( A  B)  ( A  C )
A  ( B  C )  ?  ( A  B)  C
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Deriving new properties
using rules and Venn diagrams
B  ( A  C )  ( B  A)  ( B  C )
A  B  A  ( A  B)
A  B  A  C  A  (B  C)
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Discrete Structures
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Lecture 32
April 16, 2008
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Partitions of a set
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A collection of nonempty sets {A1,A2,…,An} is a
partition of the set A if and only if
1. A = A1  A2 … An
2. A1,A2,…,An are mutually disjoint
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An infinite set can be partitioned. The partitions can
be infinite, or can be finite.
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Proofs about power sets
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Prove that
( sets A,B)[A  B 
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P(A)  P(B)]
Prove that
( sets A)[n(A) = k  n(P(A)) = 2k ]
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Russell’s paradox
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A set can be an element or member of itself.
Consider the set S = {A | A is a set and A  A}
– Is S an element of itself?
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The halting problem
Is there a computer program (suppose it’s called Halt)
which will read as input any program pgm (plus some input
for that program), and be able to determine whether pgm
will eventually halt or loop infinitely when run on that input?
What if we write a program which just runs another
program on some input and sees whether it halts, and
prints the result?
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Discrete Structures
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Lecture 33
April 18, 2008
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Proof for the halting problem
Suppose there is a program Halt(pgm, input) which can
determine whether any program pgm will halt when run on the
input data input, or loop infinitely.
Create a new program Test (which also reads another
program as input) as follows:
void Test(pgm) {
if Halt(pgm, pgm) prints that pgm halts
then while (true) ; // infinite loop- never quit
else // Halt(pgm, pgm) prints that pgm
// loops infinitely on input
exit
}
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What if Test is run on itself?
Now run Test(Test):
– if Test(Test) halts then Halt(Test, Test) will print that Test will
halt, in which case Test(Test) loops forever
– if Test(Test) loops forever then Halt(Test, Test) will print that
Test will loop infinitely, in which case Test(Test) quits
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