Sets I
CS2100
Ross Whitaker
University of Utah
Brain Teaser
Sets
•  A set is a collection of objects
•  Basic operation:
 
 
 
 
x∈A
Predicate (evaluated as true or false)
“x is an element of A”, “x is in A”
Be careful
•  Is “x” a variable with domain from another set?
•  E.g. or does “x” refer to the letter?
 
=> Know what symbols are variables and their
domains
Some Useful Sets
•  Z – set of integers
 
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
•  N – Natural numbers
 
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N = {0, 1, 2, 3, …}
Nonnegative integers
•  Q – Rational numbers
•  R – Real numbers
•  Modify with superscripts
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Z+, Z-, Z≥0, R+, …
More Definitions
•  A⊆B
 
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∀x, x∈A->x∈B
“A is a subset of B”
•  A=B
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∀x, x∈A<->x∈B
(A⊆B)∧(B⊆A)
•  Empty set
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Set containing no elements
denoted {}, or ∅
•  U – Universal set or “universe”
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Set containing all possible elements (in context)
Every set (in context) is a subset of U
First Proof
•  Proper subset
 
A⊂B <-> (A⊆B)∧(∃x∈B, x∉A)
•  Prove that (A⊂B)->¬(B⊆A)
Some Set Notation
•  “Call the set of all even, natural
numbers E”
•  E={x∈N: x is even}
 
E={x∈N: ∃m, x=2m}
•  Set builder notation
 
 
set_name={domain: predicate}
every element of the domain that satifies
the predicate is in the set.
Even More Definitions
•  For a universe: U
•  A∩B
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A∩B={x∈U: (x∈A)∧(x∈B)}
The intersection of sets A and B
•  A∪B
 
 
A∩B={x∈U: (x∈A)∨(x∈B)}
The union of sets A and B
•  A-B
 
A-B={x∈U: (x∈A)∧(x∉B)}
More Definitions
(Will It Ever End?)
• 
• 
• 
• 
∅ is a subset of every set
Every set is a subset of U
Two sets are disjoint iff A∩B=∅
A’=U-A
 
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The complement of a set A, denoted A’
Elements of the universal set that are not
in A
Basic Properties of Set Operators
Venn Diagrams
U
A
B
C
Hints on Venn Diagrams
•  Remember the Universe
•  Always show the general case
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I.e. any two sets have some regions that
overlap and some that do not
Unless there is specific reason not to
•  I.e. premises such as
 
 
A∩B=∅
A⊂B
Examples
•  Use Venn diagrams to illustrate:
 
A∩(B∪C)=(A∪B)∩(A∪C)
•  Use Venn diagrams to illustrate:  
A-(B∪C)=(A-B)∩(A-C)
Size of A Set
•  If A has a finite number of elements,
then n(A) indicates the number of
elements in A
 
 
Often denoted A
Cardinality of A
Sets as Data Containers
•  Elements have an atomic, binary operation
 
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“=” – how you define it is up to you
defined for any two elements of the “universe”
Predicate
•  The “=” returns false for any two members of a set  
I.e. sets don’t allow repeat elements
•  x∈A
 
Returns true if “x=” evaluates to true an element of A (loop
through A)
•  Two sets are the same if their members are the same
 
No sense of order
•  Data operations:
 
Can build sets by appending elements one at a time
•  Union of set and new element
•  Check for repeats
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Size is the number of data objects
Universe defines the data type
Inclusion-Exclusion Principle
•  n(A∪B) = n(A) + n(B) - n(A∩B)
•  n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) – n(A∩C) - n(B∩C)
Practice
•  Let T={x∈N, x≤1000}
•  Use the inclusion-exclusion principle to find the
size of the set:
 
G={n∈T, (2n)∨(3n)}
•  (2n)=?
 
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each element is 2k, for k=1…q
max value for q is 1000÷2=500
•  (3n)=1000÷3=333
•  For the intersection
 
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(2n)∧(3n) which is the same as (6n)
This size is 1000÷6=166
•  G = 500 + 333 – 166 = 667
Practice
•  Let T={x∈N, x≤1000}
•  Use the inclusion-exclusion principle to
find the size of the set:
 
{n∈T, (2n)∨(3n)∨(5n)}