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Discrete Mathematics
BCSC1010
Module 1
Dr. Praveen Mittal
Sets(Lecture1)
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Introduction
Discrete mathematics is the study of discrete
objects
Discrete means ‘distinct or not connected’
Modern mathematics deals with sets not
numbers
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Sets
A set is an unordered collection of objects
The objects in a set have similar properties
The objects in a set are called the elements or
members of the set
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Sets
Lowercase letters are usually used to denote
elements of sets
a,b,c,….,x,y,z
Sets are denoted by uppercase letters
A,B,C,…..,X,Y,Z
x ϵ A denotes that ‘x’ is an element of the set A
x A denotes that ‘x’ is not an element of the set A
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Some Standard Sets
N = {1,2,3, ... }, the set of natural numbers
Z = { ... , -2, -1,0, 1,2, ... }, the set of integers
Z+ = {I, 2, 3, ... }, the set of positive integers
Q = {p/q | p ϵ Z, q ϵ Z, and q ≠0}, the set of
rational numbers
R= the set of real numbers
C=the set of all complex numbers
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Representation of Sets
Roaster or Tabular form
Rule Method or Set Builder form
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Roaster or Tabular form
The notation {a, b, c, d} represents the set with
the four elements a, b, c, and d
Example:
The set V of all vowels in the English alphabet
can be written as
V = {a, e, i, o, u}
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Roaster or Tabular form
(Examples)
The set O of odd positive integers less
than 10 can be expressed by
O= {1, 3, 5, 7, 9}
The set of positive integers less than 100
can be denoted by
{1, 2, 3, ... , 99}
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Rule Method or Set Builder form
We can also "build" a set by describing what is in it
 It says "the set of all x's, such that x is greater than 0".
 In other words any value greater than 0
 Sometimes ":" can be used instead of "|", so we can write
{ x : x > 0 }
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Rule Method or Set Builder form
Ref: https://www.mathsisfun.com/sets/set-builder-notation.html
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Rule Method or Set Builder form
Ref: https://www.mathsisfun.com/sets/set-builder-notation.html
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Types of Set
Finite Set
Infinite Set
Null Set
Singleton Set
Subset
Super Set
Proper Subset
Equal Set
Universal Set
Power Set
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Empty Set
 A set with no elements is called the null or empty set. It
is represented by the symbol { } or Ø .
Examples
 the set of months with 32 days
 The set of dogs with six legs
 The set of squares with 5 sides
 The set of cars with 20 doors
 The set of integers which are both even and odd
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Empty Set (Example)
A={x: x is natural number less than 1}
Since there is no such natural number exists.
Thus,
A={} or Ø
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Finite Set
Finite sets are the sets having a finite/countable
number of elements.
Examples
A set of English alphabets; E={a,b,c,…..,z}
A set of natural numbers less than 6;
A={1,2,3,4,5}
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Infinite Set
If the number of elements in a set is not
countable/infinite, then it is called an infinite set.
Examples
 Set of all natural numbers
{1,2,3,…}
 A set of all points on a line
 The set of leaves on a tree
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Singleton Set
A set which contains only one element
is called a singleton set.
For example:
B = {x : x is a whole number, x < 1}
This set contains only one element 0
and is a singleton set.
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Singleton Set
 Let A = {x : x N and x² = 4}
Here A is a singleton set because there is only
one element 2 whose square is 4.
 Let B = {x : x is an even prime number}
Here B is a singleton set because there is only
one prime number which is even, i.e., 2.
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Subset
 If A and B are two sets, and every element of set A is also
an element of set B, then A is called a subset of B and we
write it as A B
 Symbol ‘ ’ is used to denote ‘is a subset of’ or ‘is
contained in’.
 Every set is a subset of itself
 Null set or is a subset of every set.
 Number of subsets of a set= where n is total number
of elements in a set
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Subset (Examples)
 Let A = {2, 4, 6}
B = {6, 4, 8, 2}
Here A is a subset of B, since all the elements of set A
are contained in set B.
 Vowels in English
 Population of Mathura in Uttar Pradesh
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Super Set
 Whenever a set A is a subset of set B, we say the B is
a superset of A and we write, B A.
 Symbol is used to denote ‘is a super set of’
 Example
A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A B i.e., A is a subset of B but B A i.e., B is a
super set of A
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Proper Subset
If A and B are two sets, then A is called the
proper subset of B if A B but A ≠ B
It is denoted by A B
The symbol ‘ ’ is used to denote proper subset
No set is a proper subset of itself
Null set or is a proper subset of every set
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Proper Subset (Example)
A = {1, 2, 3, 4}
B = {1, 2, 3, 4, 5}
We observe that, all the elements of A are
present in B but the element ‘5’ of B is not
present in A.
So, we say that A is a proper subset of B.
Symbolically, we write it as A B
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Equal set
Two sets are equal if and only if they have the same
elements
That is, if A and B are sets, then A and B are equal if
and only if x(x ϵ A ↔ x ϵ B). It is denoted by A =
B.
If A B and B A then A=B
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Equal set (Examples)
The sets {1, 3, 5} and {3, 5, 1} are equal
because they have the same elements
Set {1, 3, 3, 3, 5, 5, 5, 5} is same as the
set {1, 3, 5} because they have the same
elements.
Order of elements is meaningless
It does not matter how often the same
element is listed.
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Universal Set
A set which contains all the elements of other
given sets is called a universal set.
The symbol for denoting a universal set is .
Example:
 If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A U, B U, C U and U A, U B, U C]
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Power Set
 The collection of all subsets of set A is called the
power set of set A.
 It is denoted by P(A).
Example
If A = {p, q} then all the subsets of A will be
P(A) = { , {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = where m is
the number of elements in set A.
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Find the power set of {0,1,2} set
Solution:
 The power set P({0, I, 2}) is the set of all subsets
of
{0, 1, 2}. Hence,
P({0,1,2})={Φ,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}
 Note that the empty set and the set itself
are members of this set of subsets.
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
Cardinality of a Set
Cardinality of a set S, denoted by |S| is the
number of elements of the set. The number
is also referred as the cardinal number. If a
set has an infinite number of elements, its
cardinality is ∞.
Example −
| {1,4,3,5} | = 4,
| {1,2,3,4,5,…} | = ∞
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
In next lecture we will discuss…
Operations on sets
Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura

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SetTheory(Dr. Praveen Mittal).pdf

  • 1. Discrete Mathematics BCSC1010 Module 1 Dr. Praveen Mittal Sets(Lecture1) Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 2. Introduction Discrete mathematics is the study of discrete objects Discrete means ‘distinct or not connected’ Modern mathematics deals with sets not numbers Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 3. Sets A set is an unordered collection of objects The objects in a set have similar properties The objects in a set are called the elements or members of the set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 4. Sets Lowercase letters are usually used to denote elements of sets a,b,c,….,x,y,z Sets are denoted by uppercase letters A,B,C,…..,X,Y,Z x ϵ A denotes that ‘x’ is an element of the set A x A denotes that ‘x’ is not an element of the set A Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 5. Some Standard Sets N = {1,2,3, ... }, the set of natural numbers Z = { ... , -2, -1,0, 1,2, ... }, the set of integers Z+ = {I, 2, 3, ... }, the set of positive integers Q = {p/q | p ϵ Z, q ϵ Z, and q ≠0}, the set of rational numbers R= the set of real numbers C=the set of all complex numbers Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 6. Representation of Sets Roaster or Tabular form Rule Method or Set Builder form Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 7. Roaster or Tabular form The notation {a, b, c, d} represents the set with the four elements a, b, c, and d Example: The set V of all vowels in the English alphabet can be written as V = {a, e, i, o, u} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 8. Roaster or Tabular form (Examples) The set O of odd positive integers less than 10 can be expressed by O= {1, 3, 5, 7, 9} The set of positive integers less than 100 can be denoted by {1, 2, 3, ... , 99} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 9. Rule Method or Set Builder form We can also "build" a set by describing what is in it  It says "the set of all x's, such that x is greater than 0".  In other words any value greater than 0  Sometimes ":" can be used instead of "|", so we can write { x : x > 0 } Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 10. Rule Method or Set Builder form Ref: https://www.mathsisfun.com/sets/set-builder-notation.html Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 11. Rule Method or Set Builder form Ref: https://www.mathsisfun.com/sets/set-builder-notation.html Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 12. Types of Set Finite Set Infinite Set Null Set Singleton Set Subset Super Set Proper Subset Equal Set Universal Set Power Set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 13. Empty Set  A set with no elements is called the null or empty set. It is represented by the symbol { } or Ø . Examples  the set of months with 32 days  The set of dogs with six legs  The set of squares with 5 sides  The set of cars with 20 doors  The set of integers which are both even and odd Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 14. Empty Set (Example) A={x: x is natural number less than 1} Since there is no such natural number exists. Thus, A={} or Ø Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 15. Finite Set Finite sets are the sets having a finite/countable number of elements. Examples A set of English alphabets; E={a,b,c,…..,z} A set of natural numbers less than 6; A={1,2,3,4,5} Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 16. Infinite Set If the number of elements in a set is not countable/infinite, then it is called an infinite set. Examples  Set of all natural numbers {1,2,3,…}  A set of all points on a line  The set of leaves on a tree Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 17. Singleton Set A set which contains only one element is called a singleton set. For example: B = {x : x is a whole number, x < 1} This set contains only one element 0 and is a singleton set. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 18. Singleton Set  Let A = {x : x N and x² = 4} Here A is a singleton set because there is only one element 2 whose square is 4.  Let B = {x : x is an even prime number} Here B is a singleton set because there is only one prime number which is even, i.e., 2. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 19. Subset  If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A B  Symbol ‘ ’ is used to denote ‘is a subset of’ or ‘is contained in’.  Every set is a subset of itself  Null set or is a subset of every set.  Number of subsets of a set= where n is total number of elements in a set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 20. Subset (Examples)  Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset of B, since all the elements of set A are contained in set B.  Vowels in English  Population of Mathura in Uttar Pradesh Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 21. Super Set  Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B A.  Symbol is used to denote ‘is a super set of’  Example A = {a, e, i, o, u} B = {a, b, c, ............., z} Here A B i.e., A is a subset of B but B A i.e., B is a super set of A Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 22. Proper Subset If A and B are two sets, then A is called the proper subset of B if A B but A ≠ B It is denoted by A B The symbol ‘ ’ is used to denote proper subset No set is a proper subset of itself Null set or is a proper subset of every set Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 23. Proper Subset (Example) A = {1, 2, 3, 4} B = {1, 2, 3, 4, 5} We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A. So, we say that A is a proper subset of B. Symbolically, we write it as A B Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 24. Equal set Two sets are equal if and only if they have the same elements That is, if A and B are sets, then A and B are equal if and only if x(x ϵ A ↔ x ϵ B). It is denoted by A = B. If A B and B A then A=B Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 25. Equal set (Examples) The sets {1, 3, 5} and {3, 5, 1} are equal because they have the same elements Set {1, 3, 3, 3, 5, 5, 5, 5} is same as the set {1, 3, 5} because they have the same elements. Order of elements is meaningless It does not matter how often the same element is listed. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 26. Universal Set A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is . Example:  If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7} then U = {1, 2, 3, 4, 5, 7} [Here A U, B U, C U and U A, U B, U C] Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 27. Power Set  The collection of all subsets of set A is called the power set of set A.  It is denoted by P(A). Example If A = {p, q} then all the subsets of A will be P(A) = { , {p}, {q}, {p, q}} Number of elements of P(A) = n[P(A)] = where m is the number of elements in set A. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 28. Find the power set of {0,1,2} set Solution:  The power set P({0, I, 2}) is the set of all subsets of {0, 1, 2}. Hence, P({0,1,2})={Φ,{0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}  Note that the empty set and the set itself are members of this set of subsets. Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 29. Cardinality of a Set Cardinality of a set S, denoted by |S| is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Example − | {1,4,3,5} | = 4, | {1,2,3,4,5,…} | = ∞ Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura
  • 30. In next lecture we will discuss… Operations on sets Lecture Notes by Dr. Praveen Mittal, GLA University , Mathura