- A set is a collection of distinct objects where order and repetition do not matter. Basic set operations include union, intersection, and difference. - A set can be finite or infinite. Infinite sets can be countable, where elements can be paired with natural numbers, or uncountable. The set of real numbers is uncountable. - Key relationships between sets include subset, proper subset, power set, membership (∈), and equality determined by mutual inclusion. Venn diagrams can visualize set relationships.

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CS201: Data Structures and
Discrete Mathematics I
Basic Set Theory

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Sets
• A set is a collection of distinct objects.
• For example (let A denote a set):
A = {apple, orange, grape}
A = {1, 2, 3, 4, 5}
A = {1, b, c, d, e, f}
A = {(1, 2), (3, 4), (9, 10)}
A = {<1, 2, 3>, <3, 4, 5>, <6, 7, 8>}
A = a collection of anything that is meaningful.

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Members and Equality of Sets
• The objects that make up a set are called
members or elements of the set.
• Two sets are equal iff they have the same
members.
– That is, a set is completely determined by its
members.
– Order and repetition do not matter in a set.

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Set notations
• The notation of {...} describes a set. Each
member or element is separated by a comma.
– E.g., S = {apple, pear, grape}
– S is a set
– The members of S are: apple, pear, grape
• Order and repetition do not matter in a set.
• The following expressions are equivalent:
– {1, 3, 9}
– {9, 1, 3}
– {1, 1, 3, 3, 9}

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The membership symbol  and the
empty set 
• The fact that x is a member of a set S can be
expressed as
– x  S
– Reads:
• x is in S, or
• x is a member of S, or
• X belongs to S
• An example, S = {1, 2, 3}, 1  S, 2  S, 3  S
• The negation of  is written as  (is not in).
• The empty set is a set without any element
– Denoted by {} or 
– For any object x, x  

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Defining a Set by membership
properties
• Notation
o S = {x  T | P(x)} (or S = {x | x  T and P(x)})
o The members of S are members of an already
known set T that satisfy property P.
• An example:
o Let Z be the set of integers
o Let Z+ be the set of positive integers.
o Z+ = {x  Z | x > 0}

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Sets of numbers
• Z = The set of all integers
Z = {…, -2, -1, 0, 1, 2, …}
• Z+ = The set of positive integers
Z+ = {1, 2, 3…} = {x | x  Z and x > 0} = {x  Z | x > 0}
• Z- = The set of negative integers
Z- = {…, -3, -2, -1} = {-1, -2, -3…} = {x  Z | x < 0}
• R = The set of all real numbers
• Q = the set of all rational numbers
Q = {x  R | x = p/q and p, q  Z and q  0}
• We can use “;” to replace “and”

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Subsets
• A is a subset () of B, or B is a superset of
A iff every member of A is a member of B.
o A  B iff for all x if x  A, then x  B
• An example:
o (-2, 0, 6}  {-3, -2, -1, 0, 1, 3, 6}
• Negation: A is not a subset of B or B is not
a superset of A iff there is a member of A
that is not a member of B
o A  B iff there exist x, x  A, x  B

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Obvious subsets
– S  S
–   S

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Proper subsets
• A is a proper subset () of B, or B is a
proper superset of A iff A is a subset of B
and A is not equal to B.
o A  B iff A  B and A  B
• Examples:
o {1, 2, 3}  {1, 2, 3, 4, 5}
o Z+  Z  Q  R
o If S   then   S

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Power sets
• The set of all subsets of a set is called the power
set of the set
• The power set of S is denoted by P(S).
• Example:
– P() ={}
– P({1, 2}) = {, {1}, {2}, {1, 2}}
– P(S) = {, …, S}
– What is P({1, 2, 3})?
• How many elements does the power set of S
have? Assume S has n elements. 2 ^ n

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 and  are different
• Examples:
1  {1} is true
1  {1} is false
{1}  {1} is true
• Which of the following statement is true?
S  P(S)
S  P(S)
The 1st is true

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Mutual inclusion and set equality
• Sets A and B have the same members iff
they mutually include
– A  B and B  A
• That is, A = B iff A  B and B  A
• Mutual inclusion is very useful for proving
the equality of sets
• To prove an equality, we prove two subset
relationships.

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Universal sets
• Depending on the context of discussion
– Define a set of U such that all sets of interest
are subsets of U.
– The set U is known as a universal set
• Examples:
– When dealing with integers, U may be Z.
– When dealing with plane geometry, U may be
the set of points in the plane

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Venn diagram
• Venn diagram is used to visualize
relationships of some sets.
• Each subset (of U, the rectangle) is
represented by a circle inside the
rectangle.

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Set operations
• Let A, B be subsets of some universal set U.
• The following set operations create new sets
from A and B.
• Union:
A  B = {x  U | x  A or x  B}
• Intersection:
A  B = {x  U | x  A and x  B}
• Difference:
A  B = A B= {x  U | x  A and x  B}
• Complement
A = U  A = {x  U | x  A}

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Set union
• An example
{1, 2, 3}  {1, 2, 4, 5} = {1, 2, 3, 4, 5}
The venn diagram
1
2
3 4
5

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Set intersection
• An example
{1, 2, 3}  {1, 2, 4, 5} = {1, 2}
The venn diagram
1
2
3 4
5

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Set difference
• An example
{1, 2, 3} - {1, 2, 4, 5} = {3}
The venn diagram
1
2
3 4
5

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Set complement
• The venn diagram

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Basic set identities (equalities)
• Commutative laws
A  B = B  A
A  B = B  A
• Associative laws
(A  B)  C = A  (B  C)
(A  B)  C = A  (B  C)
• Distributive laws
A  (B  C) = (A  B)  (A  C)
A  (B  C) = (A  B)  (A  C)

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Basic set identities (cont …)
• Identity laws
  A = A   = A
A  U = U  A = A
• Double complement law
(A’)’ = A
• Idempotent laws
A  A = A
A  A = A
• De Morgan’s laws
(A  B)’ = A’  B’
(A  B)’ = A’  B’

23. Basic set identities (cont …)
• Absorption laws
A  (A  B) = A
A  (A  B) = A
• Complement law
(U)’ = 
’ = U
• Set difference law
A – B = A  B’
• Universal bound law
A  U = U
A   = 
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Infinite sets
• In a finite set, we can always designate one element as
the first member, s1, another element as the second
member, s2 and so on. If there are k elements in the set
we can list them as
– s1, s2, …, sk
• A set that is not finite is called an infinite set.
• If a set is infinite, we may still be able to select a first
element s1, a second element s2 and so on:
– s1, s2, …
• Both above sets are countable.
• Countable does not mean we can give a total number, but
means that we can say, “here is the first one” and “here is
the second one” and so on.

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Countable sets: examples
• The set of positive integer numbers are
countable.
• The set of positive rational numbers are
countable

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Uncountable sets
• There are also some sets that are
uncountable.
– The set is so large, and there is no way to
count out the elements.
• One example: The set of real numbers
between 0 and 1 is uncountable.
• A computer can only manage finite sets.

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Summary
• Sets are extremely important for Computer
Science.
• A set is simply an unordered list of objects.
• Set operations: union, intersection,
difference.