Set theory

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Module 1:Set Theory
Prof. Shiwani Gupta

History and Applications
• The editice of modern mathematics rests on the
concept of sets.
• Georg Cantor got his pHd in number theory.
• Important language and tool for reasoning.
• General: Reasoning and programming
• Real life: cellular phone, traffic lights, search box in
eBay, stock market, to a satellite
• Computer Science: Halting problem, Turing Machine
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Introduction to Set Theory
• A set is a structure, representing a
welldefined unordered collection (group,
plurality) of zero or more distinct (different)
objects called elements or member of a set.
• Set theory deals with operations on,
relations among, and statements about sets.

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Basic notations for sets
For sets, we’ll use variables S, T, U, …
• Roster or Tabular Form orListing Method: 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.
• Rule method or 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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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?
e.g. if S = {2, 3, 5, 7, 11, 13, 17, 19}, then |S|=8.
if S={CSC1130, CSC2110, ERG2020, MAT2510},then |S|=4.
if S = {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}, then |S|=6.

Countably infinite and
Uncountably Infinite Sets
• Finite sets are countable eg. Z.
• Uncountable sets are infinite as R.
• N, Z, R are infinite sets.
• |N| is denoted as 0.
• Any set which can be put into 1-1 correspondence
with N,Z is also countably infinite or denumerable eg.
AN, set of even Integers, set of –ve Integers, set of
prime nos.
• Thus every infinite set has countable infinite subset.
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Basic Set Relations: Member of
• xS (“x is in S”) is the proposition that object x is
an element 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, Universal, Singleton
Set
•  (“null”, “the empty set”, “void set”) is the unique set
that contains no elements whatsoever.
•  = {} = {x|False}
• No matter the domain of discourse, we have the axiom
x: x.
• In any application of theory of sets, the members of all sets
under investigation usually belong to some fixed large set
called Universal Set or Universe of Discourse.
• A set having only one element is Singleton set.

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Subset and Superset Relations
• If every element in set S is element in set T, then S is called
subset of T.
• ST (“S is a subset of T”)
• We also say that S is contained in B or B contains A
• ST (“S is a superset of T”) means TS.
• ST  x (xS → xT)
• S, SS.
• Note S=T  ST ST.
• means (ST), i.e. x(xS  xT)
• Eg. S={1,2,3}, B=(1,2,3,4,5}
TS /
A B

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Sets and Subsets
set equality C D C D D C=    ( ) ( )
subsets A B x x A x B     [ ]
][
)]()([
][
BxAxx
BxAxx
BxAxxBA



Fact: If , then |A| <= |B|.

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Properties of Subsets
• If A={4, 8, 12, 16} and B={2, 4, 6, 8, 10, 12, 14, 16},
then but
• because every element in A is an element of A.
• for any A because the empty set has no elements.
• If A is the set of prime numbers and
B is the set of odd numbers, then

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Proper (Strict) Subsets & Supersets
• ST (“S is a proper subset of T”) means that ST but
. Similar for ST.
ST /
S
T
Venn
Diagram
equivalent
of ST
Example:
{1,2}  {1,2,3}
Fact: If A = B, then |A| = |B|.
Fact: If , then |A| < |B|.

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Sets and Subsets (common
notations)
(a) Z=the set of integers={0,1,-1,2,-1,3,-3,...}
(b) N=the set of nonnegative integers or natural numbers
(c) Z+=the set of positive integers
(d) Q=the set of rational numbers={a/b| a,b is integer, b not zero}
(e) Q+=the set of positive rational numbers
(f) Q*=the set of nonzero rational numbers
(g) R=the set of real numbers
(h) R+=the set of positive real numbers
(i) R*=the set of nonzero real numbers
(j) C=the set of complex numbers

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Examples of sets
The set of all polynomials with degree at most three:
{1, x, x2, x3, 2x+3x2,…}.
The set of all n-bit strings:
{000…0, 000…1, …, 111…1}
The set of all triangles without an obtuse angle:
{ , ,… }
The set of all graphs with four nodes:
{ , , , ,…}

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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|.
There are different sizes of infinite sets!

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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 P(S)={,
{1}, {2}, {3},
{1,2}, {1,3}, {2,3},
{1,2,3}}
• Note that 1  {1}  {{1}} !!!!

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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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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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• {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}
Union Examples

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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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• {a,b,c}{2,3} = ___
• {2,4,6}{3,4,5} = ______
Intersection Examples

{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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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} =
___________
• 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}
{1,4,6}

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Set Difference - Venn Diagram
• A-B is what’s left after B
“takes a bite out of A”
Set A Set B
Set
A−B
Chomp!

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Set Complements
• The universe of discourse can itself be
considered a set, call it U.
• The complement of A, written , is the
complement of A w.r.t. U, i.e., it is U−A.
• E.g., If U=N,
A
,...}7,6,4,2,1,0{}5,3{ =

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More on Set Complements
• An equivalent definition, when U is clear:
}|{ AxxA =
A
U
A

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Ring Sum (Symmetric
Difference)
• Let A and B be two non empty sets then
ring sum of A and B is set of all elements
which are either in A or in B but not in
both.
• Thus A⊕B={x | x ∈ A∪B but ∉ A∩B}
= {x|(x∈A and x∉B)or (x∈B and x ∉A)}

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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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Cartesian Product
• Let A be the set of letters, i.e. {a,b,c,…,x,y,z}.
• Let B be the set of digits, i.e. {0,1,…,9}.
AxA is just the set of strings with two letters.
BxB is just the set of strings with two digits.
AxB is the set of strings where the first character is a letter
and the second character is a digit.
Definition: Given two sets A and B, the Cartesian product A
x B is the set of all ordered pairs (a,b), where a is in A and b
is in B. (1,2) ≠ (2,1)

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Laws of set theory
• Identity: A=A AU=A
• Domination: AU=U A=
• Idempotent: AA = A = AA
• Involution:
• Commutative: AB=BA AB=BA
• Associative: A(BC)=(AB)C
A(BC)=(AB)C
AA =)(

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Laws of set theory
• Distributive: A∩(BC)=(A∩B)(A∩C)
A (BC)=(AB)(AC)
• Properties of complement:
AAc =U AAc = 
c =U Uc= 

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DeMorgan’s Law for Sets
• Exactly analogous to (and derivable from) DeMorgan’s
Law for propositions.
BABA
BABA
=
=

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Set Operations and the Laws of
Set Theory
s dual of s (sd)


U
U
 
 
Theorem (The Principle of Duality)

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Partition of sets
A collection of nonempty sets {A1, A2, …, An} is a partition
of a set A if and only if
A1, A2, …, An are mutually disjoint (or pairwise disjoint).
e.g. Let A be the set of integers.
A1 = {x A | x = 3k+1 for some integer k}
A2 = {x A | x = 3k+2 for some integer k}
A3 = {x A | x = 3k for some integer k}
Then {A1,A2,A3} is a partition of A

Partition of sets
e.g. Let A be the set of integers divisible by 6.
A1 be the set of integers divisible by 2.
A2 be the set of integers divisible by 3.
Then {A1,A2} is not a partition of A, because A1 and A2 are not disjoint,
and also A A1 A2 (so both conditions are not satisfied).
e.g. Let A be the set of integers.
Let A1 be the set of negative integers.
Let A2 be the set of positive integers.
Then {A1,A2} is not a partition of A, because A ≠A1 A2
as 0 is contained in A but not contained in A1 A2

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(Addition Principle / Sum Rule)
If sets A and B are disjoint, then
|A  B| = |A| + |B|
A B
What if A and B are not disjoint?

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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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Inclusion Exclusion (2 sets)
For two arbitrary sets A and B
|||||||| BABABA −+=
A
B

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Inclusion Exclusion (3sets)
A B
C
|A  B  C| = |A| + |B| + |C| – |A  B| – |A  C| – |B  C|
+ |A  B  C|

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Inclusion Exclusion (4 sets)
A B
C
|A  B  C  D| = |A| + |B| + |C| + |D|
– |A  B| – |A  C| – |A  D| – |B  C| – |B  D| – |C  D|
+ |A  B  C| + |A  B  D| + |A  C  D| + |B  C  D|
– |A  B  C  D |
D

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Inclusion Exclusion (n sets)
What is the inclusion-exclusion formula for the union of n
sets?

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Counting and Venn Diagrams
• In a class of 50 college freshmen, 30 are studying BASIC,
25 studying PASCAL, and 10 are studying both. How
many freshmen are studying either computer language?
| | | | | | | |A B A B A B = + − 
A B
10 1520

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