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Numeration System
Exercise
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Introduction
Comparing Sets
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Exercise
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Sets and Number Systems
The empty set
Set Operations
De Morgan’s Law
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Institute of Mathematics
University of the Philippines-Diliman
[email protected]
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Julius M. Basilla
2014 Jan 8
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Sets and Elements
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Sets - is an aggregate/collection of objects
The individual objects in a set are called the elements.
A set S must be well-defined. For any object, one can
decide whether the object is in the set of not.
Every object in a set must be unique. That is, the same
object cannot be included in a set more than once.
Examples
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Numeration System
Exercise
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Introduction
Comparing Sets
The empty set
Set Operations
De Morgan’s Law
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{Dopey, Grumpy, Doc, Bashful, Sleepy, Sneezy, Happy}
{2, 4, 6, . . . }
{January, February, March, April, May, June, July,
August, September, October, November, December}
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Describing a set
Numeration System
Exercise
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X = {Dopey, Grumpy, Doc, Bashful, Sleepy, Sneezy,
Happy}
Y = {2, 4, 6, . . . }
Z = {January, February, March, April, May, June, July,
August, September, October, November, December}
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Roster Method- the elements are listed.
Introduction
Comparing Sets
The empty set
Set Operations
De Morgan’s Law
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X = { x : x is a dwarf of Snowwhite}
Y = { x : x is an even counting number}
Z = {x: x is a month.}
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Descriptive Method- a membership criteria is given.
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Notation
Numeration System
Exercise
JBasilla
1. If an element x is an object of a set S, we write
Introduction
Comparing Sets
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The empty set
Dopey ∈ X.
2 ∈ Y , 100 ∈ Y , 1004 ∈ Y .
January ∈ Z.
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2. Example
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x ∈ S.
Set Operations
De Morgan’s Law
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x 6∈ X.
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3. Example
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If the object x is not in X we write
Rasputin 6∈ X.
3 6∈ Y , π 6∈ Y , e 6∈ Y .
Monday 6∈ Z.
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Visualizing a set
Venn Diagram or Euler Diagram
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
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The empty set
Set Operations
De Morgan’s Law
Rasputin
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Dopey
The objects that are in the set are drawn inside the
circle representing the set
The objects that are not in the set are drawn outside the
circle representing the set.
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Equality
Numeration System
Exercise
JBasilla
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Let A be the set consisting of the 5 smallest positive
integer.
Let B = {5, 3, 2, 1, 4}.
Is A = B? Yes
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Definition. Two sets are equal if they have exactly the
same elements.
Example
Introduction
Comparing Sets
The empty set
Set Operations
De Morgan’s Law
Example
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Let A be the set of all vowels in the English Alphabet
Let B be the set of all vowels in the Filipino Alphabet.
Is A = B? Yes
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Example
Let A be the set of all primary colors.
Let B be the set of all colors appearing in the Philippine
flag.
Is A = B? No
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Subsets
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Definition. A set A is a subset of B, written A ⊂ B if and
only if all the elements of A are also in B. This means
that the statement ” If x is in A then x is in B,” is true.
Example
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Numeration System
Exercise
JBasilla
Let A be the set consisting of the even positive integer
less than 6.
Let B = {5, 3, 2, 1, 4}.
Is A ⊂ B? Yes
Introduction
Comparing Sets
The empty set
Set Operations
De Morgan’s Law
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Let A be the set of all vowels in the English Alphabet
Let B be the set of all letters in the English Alphabet
Is A ⊂ B? Yes
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Example
Example
Let A be the set of all students in this class.
Let B be the set of all engineering students.
Is A ⊂ B? No
Two sets A and B are equal if A ⊂ B and B ⊂ A.
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Empty set
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
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The empty set
Set Operations
De Morgan’s Law
The set containing nothing is called an empty set.
An empty set is a subset of any set.
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There is only one empty set, called the null set.
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Intersection
Numeration System
Exercise
JBasilla
Introduction
Figure : X ∩ Y := { x ∈ X and x ∈ Y }
Comparing Sets
X
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Set Operations
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The empty set
Y
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Union
Numeration System
Exercise
JBasilla
Introduction
Figure : X ∪ Y := { x ∈ X or x ∈ Y }
Comparing Sets
De Morgan’s Law
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Set Operations
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The empty set
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Complement
Numeration System
Exercise
JBasilla
Introduction
Figure : X 0 := { x 6∈ X}
Comparing Sets
De Morgan’s Law
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Set Operations
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The empty set
X
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The case of 3 sets
Numeration System
Exercise
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Introduction
Comparing Sets
A
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Set Operations
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C
De Morgan’s Law
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The empty set
B
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The case of 3 sets
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
The empty set
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Figure : A ∩ B
De Morgan’s Law
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C
B
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The case of 3 sets
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
The empty set
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Figure : B ∩ C
De Morgan’s Law
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C
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The case of 3 sets
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
The empty set
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Figure : A ∩ C
De Morgan’s Law
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C
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The case of 3 sets
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
The empty set
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Set Operations
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Figure : A ∩ B ∩ C
De Morgan’s Law
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C
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 2
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 3
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 5
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 30
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A complex example
Numeration System
Exercise
JBasilla
X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 10
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A complex example
Numeration System
Exercise
JBasilla
X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 15
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A complex example
Numeration System
Exercise
JBasilla
X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 6
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 6 or 10
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A complex example
Numeration System
Exercise
JBasilla
X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 6 or 15
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A complex example
Numeration System
Exercise
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X
Introduction
Comparing Sets
The empty set
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Set Operations
De Morgan’s Law
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Z
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
Y
multiples of 15 or 10
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A complex example
Numeration System
Exercise
JBasilla
X
7
Comparing Sets
The empty set
De Morgan’s Law
Z
15
2
6
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10
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Introduction
Set Operations
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X : counting numbers which are multiples of 2
Y : counting numbers which are multiples of 3
Z : counting numbers which are multiples of 5
3
Y
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(A ∪ B)0 = A0 ∩ B 0
Numeration System
Exercise
JBasilla
(A ∪ B)0
Introduction
Comparing Sets
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The empty set
Set Operations
De Morgan’s Law
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A0 ∩ B 0
A
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(A ∩ B)0 = A0 ∪ B 0
Numeration System
Exercise
JBasilla
(A ∩ B)0
Introduction
Comparing Sets
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The empty set
Set Operations
De Morgan’s Law
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A0 ∪ B 0
A
B
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The real number system
Numeration System
Exercise
JBasilla
Introduction
Comparing Sets
De Morgan’s Law
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The empty set
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