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Lecture 2
Introduction To Sets
CSCI – 1900 Mathematics for Computer Science
Fall 2014
Bill Pine
Lecture Introduction
• Reading
– Rosen - Section 2.1
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Set Definition and Notation
Set Description and Membership
Power Set and Universal Set
Venn Diagrams
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Set Definition
• Set: any well-defined collection of objects
– The objects are called set members or elements
– Well-defined - membership can be verified with a
Yes/No answer
• Three ways to describe a set
– Describe in English
• S is a set containing the letters a through k, inclusively
– Roster method - enumerate using { } ‘Curly Braces’
• S = {a, b, c, d, e, f, g, h, i, j, k}
– Set builder method ; Specify common properties of the
members
• S = { x | x is a lower case letter between a and k, inclusively}
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Set Description Examples
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Not So Good
Star Wars films
S = {car, cat, C++, Java}
{a,e,i,o,u,y}
The 8 bit ASCII
character set
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Good SciFi Films
S = { 1, car, cat, 2.03, …}
a,e,i,o,u & sometimes y
The capital letters of the
alphabet
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Finite Set Examples
• Coins
– C = {Penney, Nickel, Dime, Quarter,
Fifty-Cent, Dollar}
• Data types
– D = {Text, Integer, Real Number}
• A special set is the empty set, denoted by
–Ø
–{ }
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Infinite Set Examples
• The set of all integers Z
– Z = { …, -3, - 2, -1, 0, 1, 2, 3, …}
• The set of positive Integers Z + (Counting numbers)
– Z + = { 1, 2, 3, …}
• The set of whole numbers W
– W = { 0, 1, 2, 3, …}
• The Real Numbers R
– Any decimal number
• The Rational Numbers Q
– Any number that can be written as a ratio of two integers
• Example of a number that is in R but not in Q ?
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Additional Set Description
• The set of even numbers E
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E is the set containing … -8, -6, -4, -2, 0, 2, 4, 6, 8, …
E = any x that is 2 * some integer
E = Set of all x | x = 2*y where y is an integer
E ={ x | x = 2*y where y is an integer }
E = { x | x = 2*y where y is in Z }
E = { x | x = 2*y where y  Z }
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Set Membership
• x is an element of A is written x  A
– Means that the object x is in the set A
• x is not an element of A is written x  A
• Given: S={1, -5, 9} and Z+ the positive
Integers
– 1S
– -5  S
– 2S
1  Z+
-5  Z+
2  Z+
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Set Ordering and Duplicates
• Order of elements does not matter
– {1, 6, 9} = {1, 9, 6} = {6, 9, 1}
• Repeated elements do not matter
– {1, 1, 1, 1, 2, 3} = {1, 2, 3} = {1, 2, 2, 3}
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Set Equality
• Two sets are equal if and only if they have
the same elements
– S1 = {1, 6, 9}
– S2 = {1, 9, 6}
– S3 = {1, 6, 9, 6}
• S1 = S2 - same elements just reordered
• S2 = S3 - remember duplicates do not
change the set
• Since S1= S2 and S2 = S3 then S1=S3
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Subsets
• A is a subset of B, if and only if every
element of set A is an element of set B
– Denoted A  B
• Examples
– {Kirk, Spock}  {Kirk, Spock, Uhura}
– {Kirk, Spock}  {Kirk, Spock}
• For any set S, S  S is always true
– {Kirk, Sulu}  {Kirk, Spock, Uhura}
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Proper Subsets
• If every element of set A is an element of set B,
AND A≠B then A is a proper subset of B,
denoted A  B
• Examples
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{1,2}  {1,2,3}
{2}  {1,2,3}
{3,3,3,1}  {1,2,3}
{1,2,3}  {1,2,3}
• But {1,2,3}  {1,2,3}
– {2,3,1}  {1,2,3}
• But {2,3,1}  {1,2,3}
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Membership and Subset Exercise
Given: D = { 1, 2, {1}, {1,3}}
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Is 1  D ?
Is 3  D ?
Is 1  D?
Is {1}  D ?
Is {2}  D ?
Is {1}  D?
Is {1}  D?
Is {3}  D?
Is { {1} }  D ?
Is { {1,2} }  D ?
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Subsets and Equality
• Given: Two sets A and B
– If you know that A  B and B  A then you can
conclude that A = B
– If A  B then it must be true that B  A
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Power Set
• The power set P of a set S is a set containing
every possible unique subset of S
– Written as P(S)
• P(S) always includes
– S itself
– The empty set 
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Power Set Example
• Given: S = {x,y,z}
• P(S) = {, {x}, {y}, {z}, {x,y}, {y,z},
{x,z}, {x,y,z} }
• Nota Bene
– If there are n elements in a set S then there are
2𝑛 elements in the power set P(S)
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Set Size
• The cardinality of set S, denoted |S|, is the number
distinct elements of S.
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if S = {1,3,4,1}, then |S|=3
|{1,3,3,4,4,1}| = 3
|{2, 3, {2}, 5} | = 4
|{ 2, 3, {2,3}, 5, { 2,{2,5} } }| = 5
|Z | = ∞
|Ø| = 0
• A set is finite if it contains exactly n elements
– Otherwise the set is infinite
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Universal Set
• There is no largest set containing everything
• We will use a (different) Universal Set, U,
for each discussion
– It is the set of all possible elements of the type
we want to discuss, for each particular problem
• For an example involving even and odd
integers we might say U = Z
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Venn Diagrams
– A graphic way to show sets and subsets, developed by
John Venn in the 1880’s
– A set is shown as a Circle or Ellipse, and the Universal
set as a rectangle or square
– This shows that S1  Z, and if x  S1 then x  Z
U=Z
S1 = Integers divisible by 2
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Venn Diagrams: Subsets
U=Z
S1 = Integers divisible by 2
S2 = Integers divisible by 4
This shows that
S1  Z and S2  Z and S2  S1
If x  S2 then x S1, if x  S1 then x  Z, if x  S2 then x  Z
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Venn Diagrams: Subsets 2
U=Z
S1 = Integers divisible by 2
S3 = Integers
divisible by 5
This shows that S1  Z and S3 
Z, if x  S1 then x  Z, if
x  S3 then x  Z, and there exists at least one element y
such that
y  Z and y  S1 and y  S3
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Venn Diagram Exercise
• Draw a Venn Diagram representation for
the following example:
– U = { x | x  W and x < 10 }
– A= {1, 3, 5, 7, 9}
– B = { 1, 5, 7}
– C = {1, 5, 7, 8}
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Key Concepts Summary
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Definition of a set
Ways of describing a set
Power sets and the Universal set
Set Cardinality
Draw and interpret Venn Diagrams
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