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Unit I:
Part I
Set Theory
Information Technology Department
SKN-SITS,Lonavala.
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Introduction to Set Theory
• A set is a structure, representing an unordered collection
(group, plurality) of zero or more distinct (different)
objects.
• Set theory deals with operations between, relations among,
and statements about sets.
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Basic notations for sets
• For sets, we’ll use variables S, T, U, …
• We can denote a set S in writing by listing all of its
elements in curly braces:
– {a, b, c} is the set of whatever 3 objects are denoted by a, b, c.
• Set builder notation: For any proposition P(x) over any
universe of discourse, {x|P(x)} is the set of all x such that
P(x).
e.g., {x | x is an integer where x>0 and x<5 }
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Basic properties of sets
• Sets are inherently unordered:
– No matter what objects a, b, and c denote,
{a, b, c} = {a, c, b} = {b, a, c} =
{b, c, a} = {c, a, b} = {c, b, a}.
• All elements are distinct (unequal);
multiple listings make no difference!
– {a, b, c} = {a, a, b, a, b, c, c, c, c}.
– This set contains at most 3 elements!
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Definition of Set Equality
• Two sets are declared to be equal if and only if they
contain exactly the same elements.
• In particular, it does not matter how the set is defined or
denoted.
• For example: The set {1, 2, 3, 4} =
{x | x is an integer where x>0 and x<5 } =
{x | x is a positive integer whose square
is >0 and <25}
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Infinite Sets
• Conceptually, sets may be infinite (i.e., not finite,
without end, unending).
• Symbols for some special infinite sets:
N = {0, 1, 2, …} The natural numbers.
Z = {…, -2, -1, 0, 1, 2, …} The integers.
R = The “real” numbers, such as
374.1828471929498181917281943125…
• Infinite sets come in different sizes!
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Venn Diagrams
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Basic Set Relations: Member of
• xS (“x is in S”) is the proposition that object x is
an lement or member of set S.
– e.g. 3N, “a”{x | x is a letter of the alphabet}
• Can define set equality in terms of  relation:
S,T: S=T  (x: xS  xT)
“Two sets are equal iff they have all the same
members.”
• xS : (xS)
“x is not in S”
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The Empty Set
•  (“null”, “the empty set”) is the unique set
that contains no elements whatsoever.
•  = {} = {x|False}
• No matter the domain of discourse,
we have the axiom
x: x.
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Subset and Superset Relations
• ST (“S is a subset of T”) means that every
element of S is also an element of T.
• ST  x (xS  xT)
• S, SS.
• ST (“S is a superset of T”) means TS.
• Note S=T  ST ST.
• S / T means (ST), i.e. x(xS  xT)
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Proper (Strict) Subsets & Supersets
• ST (“S is a proper subset of T”) means that
ST but T / S. Similar for ST.
Example:
{1,2} 
{1,2,3}
S
T
Venn Diagram equivalent of ST
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Sets Are Objects, Too!
• The objects that are elements of a set may
themselves be sets.
• E.g. let S={x | x  {1,2,3}}
then S={,
{1}, {2}, {3},
{1,2}, {1,3}, {2,3},
{1,2,3}}
• Note that 1  {1}  {{1}} !!!!
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Cardinality and Finiteness
• |S| (read “the cardinality of S”) is a measure
of how many different elements S has.
• E.g., ||=0, |{1,2,3}| = 3, |{a,b}| = 2,
|{{1,2,3},{4,5}}| = ____
• We say S is infinite if it is not finite.
• What are some infinite sets we’ve seen?
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Cardinality and Finiteness
• “The number of elements in a set.”
• Let A be a set.
a. If A =  (the empty set), then the
cardinality of A is 0.
b. If A has exactly n elements, n a natural
number, then the cardinality of A is n. The set
A is a finite set.
c. Otherwise, A is an infinite set.
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Notation
• The cardinality of a set A is denoted by
| A |.
a. If A =  , then | A |= 0.
b. If A has exactly n elements, then | A | =
n.
• c. If A is an infinite set, then | A | = .
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Examples:
•
•
•
•
•
A = {2, 3, 5, 7, 11, 13, 17, 19}; | A | = 8
A = N (natural numbers); | N | = 
A = Q (rational numbers); | Q | = 
A = {2n | n is an integer}; | A | = 
(the set of even integers)
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DEFINITION
• Let A and B be sets. Then,
• |A| = |B| if and only if there is a one-toone correspondence between the elements
of A and the elements of B.
• Examples:
1. A = {1, 2, 3, 4, 5}
•
B = {a, e, i, o, u}
•
1 a, 2 e, 3 i, 4 o, 5 u; |B| =
5
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DEFINITION
2. A = N (the natural numbers)
•
B = {2n | n is a natural number} (the even
natural numbers)
•
n 2n is a one-to one correspondence
between A and B. Therefore, |A| = |B|; |B| =
.
3. A = N (the natural numbers)
•
C = {2n 1 | n is a natural number} (the odd
natural numbers)
• n 2n 1 is a one-to one correspondence
between
A and C. Therefore, |A| = |C|; |C|
= .
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Countable Sets
• DEFINITIONS:
1. A set S is finite if there is a one-to-one
correspondence between it and the set
•
{1, 2, 3, . . ., n} for some natural number n.
2. A set S is countably infinite if there is a one-toone correspondence between it and the natural
numbers N.
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Countable Sets
• DEFINITIONS:
3. A set S is countable if it is either finite or
countably infinite.
4. A set S is uncountable if it is not
countable.
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Examples:
1. A = {1, 2, 3, 4, 5, 6, 7},
•
 = {a, b, c, d, . . . x, y, z}
•
are finite sets; |A| = 7, | | = 26 .
2. N (the natural numbers), Z (the integers),
and Q (the rational numbers) are
countably infnite sets;
that is, |Q| = |Z| = |N|.
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Some Facts:
1. A set S is finite if and only if for any
proper subset A  S, |A| < |S|; that is,
“proper subsets of a finite set have fewer
elements.”
2. Suppose that A and B are infinite sets
and A  B. If B is countably infinite then
A is countably infinite and |A| = |B|.
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Some Facts:
3. Every subset of a countable set is
countable.
4. If A and B are countable sets, then A 
B is a countable set.
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The Power Set Operation
• The power set P(S) of a set S is the set of all
subsets of S. P(S) = {x | xS}.
• E.g. P({a,b}) = {, {a}, {b}, {a,b}}.
• Sometimes P(S) is written 2S.
Note that for finite S, |P(S)| = 2|S|.
• It turns out that |P(N)| > |N|.
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Ordered n-tuples
• For nN, an ordered n-tuple or a sequence of
length n is written (a1, a2, …, an). The first
element is a1, etc.
• These are like sets, except that duplicates matter,
and the order makes a difference.
• Note (1, 2)  (2, 1)  (2, 1, 1).
• Empty sequence, singlets, pairs, triples,
quadruples, quintuples, …, n-tuples.
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Cartesian Products of Sets
• For sets A, B, their Cartesian product
AB : {(a, b) | aA  bB }.
• E.g. {a,b}{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• Note that for finite A, B, |AB|=|A||B|.
• Note that the Cartesian product is not
commutative: AB: AB =BA.
• Extends to A1  A2  …  An...
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The Union Operator
• For sets A, B, their union AB is the set
containing all elements that are either in A, or
(“”) in B (or, of course, in both).
• Formally, A,B: AB = {x | xA  xB}.
• Note that AB contains all the elements of A and
it contains all the elements of B:
A, B: (AB  A)  (AB  B)
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Union Examples
• {a,b,c}{2,3} = {a,b,c,2,3}
• {2,3,5}{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}
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The Intersection Operator
• For sets A, B, their intersection AB is the set
containing all elements that are simultaneously in
A and (“”) in B.
• Formally, A,B: AB{x | xA  xB}.
• Note that AB is a subset of A and it is a subset of
B:
A, B: (AB  A)  (AB  B)
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Intersection Examples
• {a,b,c}{2,3} = ___

• {2,4,6}{3,4,5} = ______
{4}
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Disjointedness
• Two sets A, B are called
disjoint (i.e., unjoined)
iff their intersection is
empty. (AB=)
• Example: the set of even
integers is disjoint with
the set of odd integers.
Help, I’ve
been
disjointed!
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Inclusion-Exclusion Principle
• How many elements are in AB?
|AB| = |A|  |B|  |AB|
• Example:
{2,3,5}{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}
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Set Difference
• For sets A, B, the difference of A and B,
written AB, is the set of all elements that
are in A but not B.
• A  B : x  xA  xB
 x   xA  xB  
• Also called:
The complement of B with respect to A.
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Set Difference Examples
• {1,2,3,4,5,6}  {2,3,5,7,9,11} =
___________
{1,4,6}
• Z  N  {… , -1, 0, 1, 2, … }  {0, 1, … }
= {x | x is an integer but not a nat. #}
= {x | x is a negative integer}
= {… , -3, -2, -1}
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Set Difference - Venn Diagram
• A-B is what’s left after B
“takes a bite out of A”
Chomp!
Set
AB
Set A
Set B
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Set Complements
• The universe of discourse can itself be
considered a set, call it U.
• The complement of A, written A, is the
complement of A w.r.t. U, i.e., it is UA.
• E.g., If U=N,
{3,5}  {0,1,2,4,6,7,...}
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More on Set Complements
• An equivalent definition, when U is clear:
A  {x | x  A}
A
A
U
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Set Identities
•
•
•
•
•
•
Identity:
A=A AU=A
Domination: AU=U A=
Idempotent: AA = A = AA
Double complement: ( A )  A
Commutative: AB=BA AB=BA
Associative: A(BC)=(AB)C
A(BC)=(AB)C
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DeMorgan’s Law for Sets
• Exactly analogous to (and derivable from)
DeMorgan’s Law for propositions.
A B  A  B
A B  A  B
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Proving Set Identities
To prove statements about sets, of the form
E1 = E2 (where Es are set expressions), here
are three useful techniques:
• Prove E1  E2 and E2  E1 separately.
• Use logical equivalences.
• Use a membership table.
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Method 1: Mutual subsets
Example: Show A(BC)=(AB)(AC).
• Show A(BC)(AB)(AC).
– Assume xA(BC), & show x(AB)(AC).
– We know that xA, and either xB or xC.
• Case 1: xB. Then xAB, so x(AB)(AC).
• Case 2: xC. Then xAC , so x(AB)(AC).
– Therefore, x(AB)(AC).
– Therefore, A(BC)(AB)(AC).
• Show (AB)(AC)  A(BC). …
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Method 3: Membership Tables
• Just like truth tables for propositional logic.
• Columns for different set expressions.
• Rows for all combinations of memberships
in constituent sets.
• Use “1” to indicate membership in the
derived set, “0” for non-membership.
• Prove equivalence with identical columns.
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Membership Table Example
Prove (AB)B = AB.
A
0
0
1
1
B AB (AB)B AB
0
0
0
0
1
1
0
0
0
1
1
1
1
1
0
0
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Membership Table Exercise
Prove (AB)C = (AC)(BC).
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C A B ( A  B)  C A  C
0
1
0
1
0
1
0
1
BC
(AC)(BC)
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Generalized Union
• Binary union operator: AB
• n-ary union:
AA2…An : ((…((A1 A2) …) An)
(grouping & order is irrelevant)
n
• “Big U” notation:
A

i
i 1
• Or for infinite sets of sets:
A
A X
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Generalized Intersection
• Binary intersection operator: AB
• n-ary intersection:
AA2…An((…((A1A2)…)An)
(grouping & order is irrelevant)
n
• “Big Arch” notation:
A

i 1
• Or for infinite sets of sets:
i
A
A X
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Multisets
• A multiset is a set of elements, each of
which has a multiplicity
• The size of the multiset is the sum of the
multiplicities of all the elements
• Example:
• {X, Y, Z} with m(X)=0 m(Y)=3,
m(Z)=2
• Unary visualization: {Y, Y, Y, Z, Z}
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Counting Multisets
• The number of ways
• to choose a multiset of
• size k from n types of elements is:
n+k-1
n-1
=
n+k-1
k
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Example : Pirates
• How many ways are there of choosing 20
pirates from a set of 5 pirates, with
repetitions allowed?
5 + 20 - 1
20
=
24
20
24
=
4
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Operations on multiset : Union
• For multisets A, B, their union AB is the
multiset containing maximum of the
multiplicities of the elements in A and in B.
• Example: A = { a, a, a, c, d, d}
B = {a, a, b, c, c}
• AB = {a, a, a, b, c, c, d, d}
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Operations on multiset :
intersection
• For multisets A, B, their intersection A  B
is the multiset such that multiplicity of an
element is equal to minimum of the
multiplicities of the elements in A and B
• Example : A = { a, a, a, c, d, d}
B = {a, a, b, c, c}
• A  B = {a, a, c}
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Operations on multiset :
difference
• For multisets A, B, their difference A - B is
the multiset such that multiplicity of an
element is equal to multiplicity of an
element in A minus multiplicity of an
element in B
• Example : A = { a, a, a, b, b, c, d, d, e}
B = {a, a, b, b, b, c, c, d, d, f}
• A - B = {a, e}
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Operations on multiset : sum
• For multisets A, B, their intersection A + B
is the multiset such that multiplicity of an
element is equal to sum of multiplicities of
the elements in A and B
• Example : A = { a, a, b, c, c}
B = {a, b, b, d}
• A  B = {a, a, a, b, b, b, c, c, d}
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