Lecture #1

Set Theory

Sets and Elements

• Definition of a set:– A set is simply a collection of objects or elements or members.

• E.g. A={1,2,3,4} describe a set A made up of four elements 1, 2, 3, and 4.

– A set is determined by its elements and not by any particular order in which the elements might be listed. Hence above mentioned set can be expressed as:• E.g. A={1,3,4,2}

– Elements making up a set as assumed to be distinct– Only one occurrence of each element even if there are

duplicates in the set:• E.g. A={1,2,3,3,4}

Specifying a Set

• Roster Method– List all the members of a set, when this is possible.– All members of the set are listed between braces– E.g.

• V = {a, e, i, o, u}.• O = {1, 3, 5, 7, 9}

– Sometimes the roster method is used to describe a set without listing all its members. Some members of the set are listed, and then ellipses (...) are used when the general pattern of the elements is obvious.

Specifying a Set

• Set Builder– Characterize all those elements in the set by

stating the property or properties they must have to be members.

– E.g. • the set O of all odd positive integers less than 10 can be

written as – O = {x | x is an odd positive integer less than 10}

• the universe as the set of positive integers as – O = {x Z∈ + | x is odd and x < 10}

Set Notation

• N = {0, 1, 2, 3,...}, the set of natural numbers• Z = {..., −2, −1, 0, 1, 2,...}, the set of integers• Z+ = {1, 2, 3,...}, the set of positive integers• Q = {p/q | p Z, q Z, and q = 0}, the set of ∈ ∈

rational numbers• R, the set of real numbers• R+, the set of positive real numbers• C, the set of complex numbers.

Intervals

• [a, b] ={x | a ≤ x ≤ b} closed interval• [a, b) ={x | a ≤ x<b}• (a, b] ={x | a<x ≤ b}• (a, b) ={x | a<x<b} open interval

Universal Set

• In any application of the theory of set, the numbers of all sets under investigation usually belong to some fixed large set called Universal Set

• It is represented by U unless otherwise specified.

Empty Set

• There is a special set that has no elements.• This set is called the empty set, or null set, and is

denoted by .∅• The empty set can also be denoted by { } (that is, we

represent the empty set with a pair of braces that encloses all the elements in this set).

• Often, a set of elements with certain properties turns out to be the null set. For instance, the set of all positive integers that are greater than their squares is the null set.

Singleton Set

• A set with one element is called a singleton set.

• A common error is to confuse the empty { } ∅has one more element than .∅

• Set with the set { }, which is a singleton ∅ ∅set.

• The single element of the set { } is the empty ∅set itself.

Venn Diagram

• Sets can be represented graphically using Venn diagrams

• In Venn diagrams– the universal set U, which contains all the objects under

consideration, is represented by a rectangle.– Inside this rectangle, circles or other geometrical figures

are used to represent sets.– Sometimes points are used to represent the particular

elements of the set.– Sometimes points are used to represent the particular

elements of the set.

Subsets• It is common to encounter situations where the elements of one set

are also the elements of a second set.• The set A is a subset of B if and only if every element of A is also an

element of B.OR

• If every element is a set A is also an element of a set B then A is called a subset of B.

• We use the notation A B to indicate that A is a subset of the set B.⊆• If at least one element of A does not belong to B, we write A ⊈ B• If A⊆B, then it is still possible that A=B.• When A⊆B, but A≠B, we say A is a proper subset of B written a A⊂B.

Subsets

• Examples– Suppose there are three sets A, B, and C as

following• A={1,3}• B={1,2,3}• C={1,3,2}

– Sets A and B both are subsets of Set C• Set A is a proper subset of Set C A⊂C• Set B is an improper subset of Set C B⊆C

Finite Sets

• A set is said to be finite if it contains exactly n elements where n is a nonnegative integer.

• Finite set have a one-to-one correspondence between the elements in the set and the element in some set n, where n is a natural number and n is cardinality of the set.

Cardinality

• It is a measure of the "number of distinct elements of the set".

• Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer,

• we say that S is a finite set and that n is the cardinality of S.

• Cardinality of S is denoted by |S|.• For example, the set A = {2, 4, 6} contains 3 distinct

elements, and therefore A has a cardinality of 3.

Infinite Sets• Definition:

– A set which is not finite, is called an infinite set.• Countable Infinite Sets

– A set is countably infinite if its elements can be put in one-to-one correspondence with the set of natural numbers. In other words, one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

– For example, the set of integers {0,1,−1,2,−2,3,−3,…} is clearly infinite. However, as suggested by the above arrangement, we can count off all the integers.

– Counting off every integer will take forever. But, if you specify any integer, say −10,234,872,306, we will get to this integer in the counting process in a finite amount of time.

Power Sets

• Many problems involve testing all combinations of elements of a set to see if they satisfy some property.

• To consider all such combinations of elements of a set S, we build a new set that has as its members all the subsets of S.

• Given a set S, the power set of S is the set of all subsets of the set S. The power set of S is denoted by P(S).

Power Sets

• Example– What is the power set of the set {0, 1, 2}?– The power set P({0, 1, 2}) is the set of all subsets

of {0, 1, 2}.– Hence, P({0, 1, 2}) = { , {0}, {1}, {2}, {0, 1}, {0, 2}, ∅

{1, 2}, {0, 1, 2}}.

Cartesian Products

• Because sets are unordered– A different structure is needed to represent

ordered collections.– This is provided by ordered n-tuples.

• Definition– Let A and B be sets. The Cartesian product of A

and B, denoted by A × B, is the set of all ordered pairs (a, b), where a A and b B.∈ ∈

– Hence, A × B = {(a, b) | a A b B}.∈ ∧ ∈

Cartesian Products

• The ordered n-tuple (a1, a2, . . . , an) is the ordered collection that has a1 as its first element, a2 as its second element, . . . , and an as its nth element.

• The Cartesian product– of the sets A1,A2, . . . , An,

– denoted by A1 × A2 × ・ ・ ・ ×An,

– is the set of ordered n-tuples (a1, a2, . . . , an),

– where ai belongs to Ai for i = 1, 2, . . . , n.

• In other words,– A1 × A2 × ・ ・ ・ ×An = {(a1, a2, . . . , an) | ai A∈ i for i = 1, 2, . . . , n}.

Cartesian Products

• Examples #1– What is the Cartesian product of A = {1, 2} and B =

{a, b, c}?– The Cartesian product A × B is

• A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}.

• Example #2– What is the Cartesian product A × B × C, where A =

{0, 1}, B = {1, 2}, and C = {0, 1, 2} ?• A × B × C = {(0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 2, 0), (0, 2, 1), (0,

2, 2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0), (1, 2, 1), (1, 2, 2)}.

Set Operations

• Union– of sets A and B contains elements in A, B or

both A ∪ B = { x | x A ∈ v x B}∈– Example• {1, 2, 3} ∪ {3, 4} = {1, 2, 3, 4}• {1, 2, 3} ∪ {1, 2, 3} = {1, 2, 3}

Set Operations

• Intersection– of sets A and B contains those elements in both A

and B– A ∩ B = { x | x A ∈ ^ x B}∈– Example• {1, 2, 3} ∩ {3, 4} = { 3 }• {1, 2, 3} ∩ {4} = ∅• {1, 2, 3} ∩ {1, 2, 3} = {1, 2, 3}

Set Operations

• Disjoint– when intersection is empty A ∩ B = .∅– Example• {1, 2 } ∩ {3, 4} = ∅

• Cardinality of union.– |A ∪ B| = |A| + |B| - |A ∩ B|

• Difference, – A - B is set containing elements in A but not in B. – A - B = { x | x A ∈ ^ x B}∉

Set Operations

• Complement, U - A or Ac

– is with respect to the universal set, U.• A = { x | x ∈ U ^ x A}∉• A = { x | x A}∉

Set Operations

• Fundamental Products– Consider n distinct sets A1, A2, A3, …, An. A

fundamental product of the sets is a set of the form

AAAA n

**

3

*

2

*

1 ...

Set Operations

• Fundamental Products– Example: Consider three sets A, B, and C. The following lists

the eight fundamental products of the three sets– P1 = A∩B∩C– P2 = A∩B∩Cc

– P3 = A∩Bc∩C– P4 = A∩Bc∩Cc

– P5 = Ac∩B∩C– P6 = Ac∩B∩Cc

– P7 = Ac∩Bc∩C– P8 = Ac∩Bc∩Cc

Set Operations

• Symmetric Difference– The symmetric difference of sets A and B, denoted

by A ⊖B, consists of those elements which belong to Set A or Set B but not to both that is• A⊖B = (A\B)∪(A∩B) or A⊖B = (A\B)∪(B\A)

– Example: Suppose A={1,2,3,4,5,6} and B={4,5,6,7,8,9}.

– Then A\B={1,2,3} and B\A={7,8,9} – A⊖B = {1,2,3,7,8,9}

Set Identities

Inclusion-Exclusion Principle

• The inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as – n(A∪B) = n(A) + n(B)– n(A∪B) = n(A) + n(B) – n(A∩B)

Partitions

• Let S be a non-empty set.• A Partition of S is a sub-division of S into subsets that

are– Non-overlapping– Non-empty

• Precisely, a partition of S is a collection {Ai} of non-empty subsets of S such that:– Each a in S belongs to one of the Ai

– The sets of {Ai} are mutually disjoint that is, if• Ai≠ Aj then Ai∩ Aj= Ø

Examples of Partition

Multisets

• Sets:– An unordered collection of distinct objects.

• Multisets:– Sets in which some elements occur more than once– A={1,1,1,2,2,3}

• Notation to represent a multiset by:– S={n1.a1,n2.a2,…,ni.ai}

– This denotes that a1 occurs n1 times

– The number ni=1,2,3,.. Are called multiplicities of the elements ni.

– A={3.1,2.2,1.3}

Union of Multisets

• The union of the multisets A and B is the multiset where the multiplicity of an element is the maximum of its multiplicities in A and B

Intersection Multisets

• The Intersection of A and B is the multiset where the multiplicity of an element the minimum of its multiplicities in A and B.

Difference of Multisets

• The difference of A and B is the multiset where the multiplicity of an element is the multiplicity of element in A less its multiplicity in B unless this difference is negative, in which case the multiplicity is zero.

Sum of Multisets

• The Sum of A and B is the multiset where the multiplicity of an element is sum of multiplicities in set A and set B denoted by A+B.

Multiset Examples

Practice Problems

• A={1,4,7,10}• B={1,2,3,4,5}• C={2,4,6,8}

• A U B

• B ∩ C

• A – B

• B – A

• Ac

• Uc

• B ∩ Ø

• B ∩ U

• B ∩ (C – A)

• U – C

• A U Ø

• A U U

• A ∩(B U C)

• (A ∩ B) – C

• (A ∩ B) U C

• (A U B) – (C – B)

{ } set (a,b) ordered pair a∈A element of

| such that A×B cartesian product x∉A not element of

A ∩ B intersection |A| cardinality A - B relative complement

A ∪ B union #A cardinality A ⊖ B symmetric difference

A ⊆ B subset aleph-null

A ⊂ B proper subset / strict subset aleph-one

A ⊄ B not subset Ø empty set

A ⊇ B superset universal set

A ⊃ B proper superset / strict superset 0

natural numbers / whole numbers  set (with zero)

A ⊅ B not superset 1

natural numbers / whole numbers  set (without zero)

2A power set integer numbers set

power set rational numbers set

A = B equality real numbers set

Ac complement   complex numbers set

A \ B relative complement A ∆ B symmetric difference