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9/1/2016
CISC 1400
Discrete Structures
Chapter 1
Sets
Fall 2016
CISC1400 Yanjun Li
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Introduction to Discrete Mathematics

Continuous mathematics



Discrete mathematics

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Math based on the continuous number
line, or the real numbers (floating point
number).
Algebra, geometry, trigonometry, and
calculus
Working with distinct values.
Sets, sequences, Logic, relations,
functions, counting and probability.
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Discrete Math for Computer Science
A
digital computer contains a
finite amount of memory and
virtual storage space,
measured in bits (binary digits).
 0/1 – one bit, 8 bits = 1 byte,
210 =1024 bytes = 1 kilobyte,
10242 bytes = 1 megabyte,
10243 bytes = 1 gigabyte,
10244 bytes = 1 terabyte
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What is Set?


A collection of objects
Example:




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Your
Your
Your
Your
Your
high school classmates.
high school friends.
family members.
MP3 playlist.
birthday gifts.
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Representation of Set

Set name and its
elements/members

Set A ={a, b, c, d, e, f}
6

elements: letter a, b, c, d, e, and f
Set B ={bob, 1,8,clown, hat}
5
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elements: bob, 1, 8, clown, hat.
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Features of Set

Unordered :



Distinct objects:

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The order in which you list the
elements in a set does not matter.
For example, {1, 2, 3} = {3, 2, 1}
The elements within a set should not
be duplicated.
For example, wrong {a, b, c, d, a}
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Elements of a Set
{bob,1,8, {1,2,3}, a,
b, {clown, bob}, hat ,
4, {{{{1}}}}, clown}
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Membership Operator

Two key symbols that we will see:
x  A means “x is an element of set A”
x  A means “x is not an element of set A”
a  {a, b, c}
1 {a, b, c}
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Cardinality of A Set

The cardinality of a set A is the
number elements in the set, denoted
as |A|.


A set could be considered as a bag.
A set can be finite and have zero, one,
two, or any finite number of elements.
For example, A = {1,2,3}, |A|=3.
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Cardinality of A Set

A set has no elements, {}, is called
as the empty set or the null set, and
is represented by Ø. i.e., |{}|=0.


A set can be infinite and comprised
of an infinite number of elements.

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An empty bag!
Example?
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Exercises
1.
2.
A = {alpha, beta, gamma}
B = {-5, 0, 5, 10}
C = {{a,{b}}, {c,d}}
D = {Ø, 10,11}
E=Ø
F = {Ø}
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3.
4.
What is |A|?
3
What is
|B|+|C|
6
What is
|D|+|E|-|A|?
3+0-3=0
What is |F|?
1
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Relations Between Sets

Two sets are equal when they have
the same elements.
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Two sets are not equal
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A = {1,2,3}, B= {1,2,3}
A = B, A = A
A = {1,2,3}, C={1,2}
A≠C
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Relations Between Sets

If elements of set A are all contained in
set B, then A is a subset of B.



A = {1,2,3}, B ={1,2,3,4}
A ⊆ B, B ⊇ A
If there exists one element of set A that is
not an element of set B, then A is not a
subset of B.


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A = {1,2,3,5}, B ={1,2,3,4}
A ⊈ B, B ⊉ A
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Relations Between Sets

If A is a subset of B and there is at least
one element in B that is not in A, then A is
a proper subset of B.


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A = {1,2,3}, B ={1,2,3,4}
A ⊂ B, B ⊃ A
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Commonly Used Sets

U: the universal set, containing all
possible elements that can be
placed into a set.

U: the set of all Fordham students
A: all freshman students,
 B: all female students,
 C: all science major students
 A ⊆ U, A ⊂ U
 B ⊆ U, B ⊂ U
 C ⊆ U, C ⊂ U

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Venn Diagram


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A rectangle
represents
universal set, U
Circles within it
represents sets
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Subset
A is a subset of B.
A⊆ B
B
A
A is a proper subset of B.
A⊂ B
A is not a subset of B.
A⊈ B
C is not a subset of B.
C⊈B
C
B B
A
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Exercise
 Find
out all subsets of set A={1,2}
Ø, {1}, {2}, {1,2}
 Find
out all subsets of set
A={a,b,c}
Ø, {a}, {b}, {c}, {a,b},{a,c},{b,c},{a,b,c}
 Find
out all proper subsets of set
A={a,b,c}
Ø, {a}, {b}, {c}, {a,b},{a,c},{b,c}
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Commonly Used Sets

ℕ: the set of natural numbers (nonnegative integers)



ℕ = {0,1,2,3,4,5,6,…}
1 ∈ ℕ, - 10 ∉ ℕ, 3.1415 ∉ ℕ.
ℤ : the set of all integers, both positive and
negative

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ℤ = {… -3, -2, -1, 0, 1, 2, 3, …}
ℕ ⊆ ℤ, ℕ ⊂ ℤ
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Commonly Used Sets

ℚ: the set of all rational numbers (i.e., all
numbers that can be written as a fraction)



ℝ: the set of all real numbers, which are
essentially all numbers that you can imagine.
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ℚ = { … 1/32, 1/31, …, 1/3, ½, … 2/32, 2/31}
π ∉ ℚ, √2 ∉ ℚ
ℚ ⊆ ℝ, ℚ ⊂ ℝ
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Variations
is the set of positive natural
numbers, { 1, 2, 3, 4, 5, …}
 ℤ- is the set of negative integers
{-1,-2,-3,…}
 ℚ>1 is the set of rational numbers that
are greater than 1
 ℝ<10 is the set of real numbers that are
smaller than 10
 ℕ+
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Set Builder Notation

To define a set A that contains all
positive even integers.
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A = {2, 4, 6, 8, 10, 12, 14, ….}
Set A is comprised of all positive even
integers.
A = {x: x is a positive even integer}
A = {x| x ∈ ℤ and x is positive and x is
even}
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Set Builder Notation

A = {x: x∈ℤ+ and x is even} is
translated to:
“All elements x such that x is an
element of the positive integers and
x is even”


x is the place holder stating that set A
will be comprised of some variable x.
Each of the following expressions is
used to enforce the constraint of x.
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Examples
{x : x  2  5}
{2.5}
{x : x  2k and k  {1,2,3}} { 2 , 4 , 6 }
x
{x : x  N and  N }
3
{ 0 , 3 , 6 , 9 ,12 ,15 ,...}
{x | x  2 y and y  Z  }
{ 2 , 4 , 6 , 8 ,10 ,12 ,...}
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Exercises

Answer the following questions:
A  { x  Q : x 2  1}


What is |A| ?
Are the following true ?
1.001  A

0.999  A
Find all elements of set B defined as
follows:
B  { x : x  N and x  4}
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True or False
1.
2.
3.
4.
5.
6.
If x ∈ A , and A ⊆ B, then x ∈ B
If A ⊆ B , and B ⊆ C , then A ⊆ C
If A ⊆ B , then |A| ≤ |B| when they
are both finite.
If |A| ≤ |B| , then A ⊆ B
{} has no subset.
Which of the following is true:
{1,2,3}  {{1,2,3}}
{1,2,3}  {{1,2,3}}
{1,2,3}  {{{1,2,3}}}
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Set Operation: Union
A B
Create a new set by combining all of
the elements of two sets, i.e.,
A  B  {x | x  A or x  B}
The part that has been
shaded is A ∪ B.
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Union Examples
A  {1,2,3,4,5}
B  {0,2,4,6,8}
C  {0,5,10,15}
D  {}
A  B  {0,1,2,3,4,5,6,8}
B  A  {0,1,2,3,4,5,6,8}
C  D  {0,5,10,15}
A C  {0,1,2,3,4,5,10,15}
D  B  {0,2,4,6,8}
( A  C )  ( D  B )  {0,1,2,3,4,5,6,8,10,15}
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Set Operation: Intersection
A B
Create a new set using the elements the
two sets have if common
A  B  {x | x  A and x  B}
The dark part is A ∩ B
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Intersection Examples
A  {1,2,3,4,5}
B  {0,2,4,6,8}
C  {0,5,10,15}
D  {}
A B 
{ 2 , 4}
B A
{2,4}
CD 
{}
AC 
{5}
DB 
{}
( A  C )  ( D  B )  {5}
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Set Operation: Difference
A B
Create a new set that includes all elements
of set A, removing those elements that are
also in set B
A  B  {x | x  A and x  B}
A
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A-B
B
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Difference Examples
A  {1,2,3,4,5}
B  {0,2,4,6,8}
C  {0,5,10,15}
D  {}
A  B  {1,3,5}
B  A  {0,6,8}
C  D  {0,5,10,15}
( A  C )  ( D  B )  {1,2,3,4}
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Set Operation: Complement


Universal set: the set that includes
everything
The difference of U and A is also called
the complement of A:
Ac  U  A  {x | x  U and x  A}
U
Colored area is
U-A
A
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Set operations example

Consider the following four subsets of the set U of all
students

A is the set of all computer science majors.
B is the set of all physics majors.
C is the set of all science majors.

D is the set of all female students.



Using set operations, describe each of the following in
terms of the sets A, B, C and D:


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Set of all male physics majors.
B ∩ Dc
Set of all students who are female or science majors.
D∪C
Set of all students not majoring in science.
Cc
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Power Set
P ( A)
is a set that consists of all possible
subsets of set A including Ø
P( A)  {x : x  A}
e.g. P({1})=?
List all subsets of {1}: {},{1}
Therefore P({1})={{},{1}}.
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Power Set Examples
A  {1,2,3}
P ( A)  {{1, 2 ,3}, {1, 2}, {1,3},
B  {a, b, c, d }
{ 2 ,3}, {1}, { 2}, {3}, {}}
{{a, b, c, d }, {a, b, c}, {a, b, d },
P ( B )  {a, c, d }, {b, c, d }, {a, b}, {a, c},
{a, d }, {b, c}, {b, d }, {c, d }, {a},
{b}, {c}, {d }, {}}
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Exercises on power set
1. A  {a}
P( A) 
2. C  {a, 1 }
P(C) 
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Cardinality of Power Set

If |A|=1, |P(A)|= ?


If |A|=2, |P(A)|=


Try P({a,b})
If set A have a certain number of subsets,
after we add one more element into A,
how many subsets A has now ?



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Try P({a})=
Every originally identified subsets are still valid
Add the new element into each of them, and
we get a new subset.
The number of subsets doubles !
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Cardinality of Power Set

Let an be the number of subsets that
a set of cardinality n has, then
a1  2
a


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n 1
We can find the closed form:
a

 2a
n
n
 2
n
A set of cardinality n has 2n subsets
If |A|=n, |P(A)|=2n
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Cartesian Product (Cross Product)
A B
Create a new set consisting of all possible
ordered pairs where the first element is
from A, and second element is from B.
A  B  {( x, y ) : x  A and y  B}
AxB≠BxA
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Example of Cartesian Product

I have two T-shirts: white, black


I have three jeans: black, blue, green


A={white shirt, black shirt}
B={black jean, blue jean, green jean}
All outfits I can make out of these ?

The set of all ordered-pairs, in the form
(T-shirt, jean)…
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Cartesian Product Examples
A  {1,2,3}
B  {a, b, c}
C  {1,5}
A  B  {(1, a ), (1, b ), (1, c ), ( 2 , a ),
( 2 , b ), ( 2 , c ), ( 3, a ), ( 3, b ), ( 3, c )}
B  A  {(a,1), (a,2), (a,3), (b,1), (b,2),
(b,3), (c,1), (c,2), (c,3)}
C  A  {(1,1), (1,2), (1,3), (5,1),
(5,2), (5,3)}
BC 
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{( a ,  1), ( a ,5), (b ,  1), (b ,5),
( c ,  1), ( c ,5)}
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Cardinality of Cartesian Product

If A has m elements, B has n
elements, how many elements does
AxB have ?




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For every element of A, we pair it with
each of the n elements in B,. to get n
ordered pairs in AxB
So we can form n*m ordered pairs this
way
So |AxB| = m*n = |A|*|B|
This is where the name Cartesian
product comes
from.
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