CMSC 250
Discrete Structures
Set Theory
Sets

Definition of a Set:
– NAME = {list or description of elements}
– Examples
 B = {1,2,3}
 C = {xZ+ | - 4 < x < 4}

Axiom of Extension
– A set of elements is completely defined by
elements, regardless of order and duplicates
– Example: {a,b}={b,a}={a,b,a}={a,b,a,b,b,a}
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Set Theory
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Notation

Sets defined by property
– C = {xZ+ | - 4 < x < 4}
– C = {1,2,3,4}

Consider elements: glue, tape, pen
– Sets are not equivalent to elements
 {glue}  glue
– Sets can be elements of other sets
 {pen, {glue, tape}}
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Set Theory
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Subset

AB  xU, xA  xB
– A is contained in B
– B contains A

AB  xU, xA  xB

Relationship between membership and subset:
– xU, xA  {x}  A

Definition of set equality:
– A = B  AB  BA
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Set Theory
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Same Set 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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Set Theory
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 Versus 

glue  {glue, tape, pen}

{glue}  {glue, tape, pen}

{glue}  {{glue}, {tape}, pen}

{glue}  {{glue}, {tape}, pen}
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Set Theory
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Sets
C = {x | x > - 4 and x < 4}
 C = {xZ+ | x > - 4 and x < 4}

– What is the first element?
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Set Theory
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Set Operations
Formal Definitions and Venn Diagrams
Union:
A  B  {x U | x  A  x  B}
Intersection:
A  B  {x U | x  A  x  B}
Complement:
c
A  A'  {x U | x  A}
Difference:
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A  B  {x U | x  A  x  B}
A  B  A  B'
Set Theory
8
Procedural Versions of Set Definitions
Let X and Y be subsets of a universal set U
and suppose x and y are elements of U.
1.
2.
3.
4.
5.
x(XY) xX or xY
x(XY) xX and xY
x(X–Y) xX and xY
xXc xX
(x,y)(XY) xX and yY
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Set Theory
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Venn Diagrams
A={1,2,5,7}; B= {1,5}; C={3,7}
 U={1,2,3,4,5,6,7}

2
1
5
A
2
B
1B
5
7
C
3
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BA
4 6
4 6
Set Theory
A
7
C
3
10
Venn Diagram for R, Z, Q
Z
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Q
Set Theory
R
11
Ordered n-Tuple


Ordered n-tuple –
(x1,x2,x3,…,xn)
takes order and multiplicity into account
– n values
– not necessarily distinct
– in the order given

(x1,x2,x3,…,xn) = (y1,y2,y3,…,yn)
 iZ1in, xi=yi

Examples
– {a,b} = {b,a}
– {(a,b)}  {(b,a)}
– Cartesian coordinate system
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Set Theory
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Cartesian Product
A  B  {(a, b) | a  A  b  B}

Example
– A = {x,y,z}
– B = {5,7}
– C = {a,b}

A  B  C  (A  B)  C
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Set Theory
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Empty Set Properties
1.
2.
3.
4.
5.
6.
7.
Ø is a subset of every set.
There is only one empty set.
The union of any set with Ø is that set.
The intersection of any set with its own
complement is Ø.
The intersection of any set with Ø is Ø.
The Cartesian Product of any set with Ø is Ø.
The complement of the universal set is Ø and
the complement of the empty set is the
universal set.
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Set Theory
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Other Definitions

Proper Subset

Disjoint Set: A and B are disjoint
A B  A B A B
A and B have no elements in
common
xU, xAxB ^ xBx
A
AB = Ø  A and B are Disjoint Sets
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Set Theory
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Partitions of a Set


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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Set Theory
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Power Sets
Power set of A =
P(A) = Set of all subsets of A
Example: A={a,b,c}
 Can also think of as a truth table …


Prove that
– A,B{sets}, AB 

P(A)  P(B)
Prove that (where n(X) means the size of set X)
– A{sets}, n(A) = k  n(P(A)) = 2k
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Set Theory
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Edwards-Venn Diagram
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Set Theory
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Properties of Sets (Theorems 5.2.1 & 5.2.2)
A B  A
A  A B
A B  B
B  A B

Inclusion

Transitivity

DeMorgan’s for Complement

Distribution of union and intersection
A BBC  AC
( 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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Set Theory
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Element Argument

Basic Method for proving that one set is a
subset of another

Let sets X and Y be given. To prove XY,
– Suppose that x is a particular but arbitrarily
chosen element of X,
– Show that x is an element of Y.
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Set Theory
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Set Equality
Two sets A and B are equal, if and only if,
they have the same exact elements.
 Expressed as:

– A=B
– A=B
– A=B
– A=B
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



AB and BA
(xA, xB)  (xB, xA)
xU, (xAxB  xBxA)
(xU, xAxB) 
(xU, xBxA)
Set Theory
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Prove A=C
A={nZ | pZ, n = 2p}
C={mZ | qZ, m = 2q-2}
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Set Theory
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Does A=D
A={xZ | pZ, x = 2p}
D={yZ | qZ, y = 3q+1}
Easy to disprove universal statements!
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Set Theory
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Prove A – B = A – (A  B)
LHS: A – B = {xU | xA  xB}
 RHS: A – (A  B)

=
=
=
=
=
=
=
=

{xU
{xU
{xU
{xU
{xU
{xU
{xU
{xU
|
|
|
|
|
|
|
|
xA  x(A  B)}
xA  x(A  B)’}
xA  x(A’  B’)}
xA  (xA’  xB’)}
(xA  xA’)  (xA  xB’)}
FALSE  (xA  xB’)}
xA  xB’}
xA  xB}
LHS = RHS
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Set Theory
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Prove A  B  A
sets A,B xU x(A  B)  xA
 Choose generic sets A, B and element xU
 Assume x(A  B)

– xA  xB (by def of intersection “”)
– xA (by conjunctive simplification)
x(A  B)  xA (by closing cond. world)
 sets A,B xU x(A  B)  xA

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Set Theory
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Using Venn Diagrams to help
find counter example
A  B  C  ?  A  B  C 
A  B  C   ?   A  B   A  C 
A  B  C   ?   A  B  C
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Set Theory
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Deriving new Properties
using rules and Venn diagrams
A  B and B  A are disjoint
A  B  A   A  B
A  B  A  C  A  B  C 
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Set Theory
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Formal Languages
 = alphabet = a finite set of symbols
 string over  =
empty (or null) string denoted as 
OR
ordered n-tuple of elements
 n = set of strings of length n
 * = set of all finite length strings


L = {s | s=aibiai for iZ≥0} = ?
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Set Theory
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