Introduction to sets | presented by sumit malviya

1. INTRODUCTION TO SETS
SETS : THEORY
PRESENTED BY : SUMIT MALVIYA

2. SET
• A specific set can be defined in two ways-
• A Set is a well defined collection of objects, called the “Elements” or
“members” of the set.
1. If there are only a few elements, they can be listed individually , by writing
them between curly braces ‘{ }’ and placing commas in between. E.g.-
{1,2,3,4,5}.
2. The second way of writing set is to use a property tha defines elements of the
set.
3. E.g.- { x | x id odd and 0 < x < 100}
• If x is an element o set A, it can be written as ‘x ∈ A’
• If x is not an element of A, it can be written as ‘x ∈ A’

3. SPECIAL SETS
• Standard notations used to define some sets:
a) N- set of all natural numbers
b) Z- set of all integers
c) Q- set of all rational numbers
d) R- set of all real numbers
e) C- set of all complex numbers

4. TYPES OF SETS
 SUBSET
 EQUAL SETS
 EMPTY SETS
 SINGLETON SET
 FINITE SET
 INFINITE SET
 CARDINAL NUMBER OF A SET
 DISJOINT SETS
 POWER SET
 UNIVERSAL SET

5. SUBSET
• If every element of a set A is also an element of set B. we say set A is a
subset of set B.
EXAMPLE:
If A={1,2,3,4,5} and B={1,2,3,4}
Then B ⊆ 𝐴

6. EQUAL SETS
• Two sets A and B are called equal if they have equal numbers and similar
types of elements
i.e. A ⊆ B
This implies, A=b
• For e.g. If A={1,3,4,5,6}
• B={4,1,5,6,3} then both set A and B are equal.

7. EMPTY SETS
 A set which does not contain any element is called as Empty set or Null or Void set.
Denoted by ∅ 𝑜𝑟 {}
 Example: (a) The set of the whole number less than 0
(b) Clearly there is no whole nimber less than 0. Therefore,it is an empty set
(c) N = [𝑥: 𝑥 𝜖 𝑁 , 3 < 𝑥 < 4
 Let A = {x : 2 < 3, x is a natural number}
Here A is an empty set because there is no natural number between 2 and 3
 Let B = {x : x is a composite number less than 4}
Here B is an empty set because there is no composite number less than 4

8. Singleton set is a set containing only one
element. The singleton set is of the form A =
{a}, and it is also called a unit set. The
singleton set has two subsets, which is the
null set, and the set itself.
SINGLETON SET

9.  Finite sets are the sets having a finite/countable number of members. Finite sets are also known as countable sets as they
can be counted. The process will run out of elements to list if the elements of this set have a finite number of members.
 Examples of finite sets:
P = { 0, 3, 6, 9, …, 99}
Q = { a : a is an integer, 1 < a < 10}
A set of all English Alphabets (because it is countable).
 Another example of a Finite set:
A set of months in a year.
M = {January, February, March, April, May, June, July, August, September, October, November, December}
n (M) = 12
It is a finite set because the number of elements is countable.
FINITE SETS

10. If a set is not finite, it is called an infinite set because the number of elements in that
set is not countable and also we cannot represent it in Roster form. Thus, infinite sets
are also known as uncountable sets.
So, the elements of an Infinite set are represented by 3 dots (ellipse) thus, it represents
the infinity of that set.
Examples of Infinite Sets
•A set of all whole numbers, W= {0, 1, 2, 3, 4,…}
•A set of all points on a line
•The set of all integers
INFINITE SETS

11. CARDINAL NUMBER OF A SET
• The number of distinct elements in a given set A is called the cardinal numberof A. It is
denoted by n(A)
• FOR EXAMPLE:
A {x : x ∈ N, x < 5 }
A= {1,2,3,4}
Therefore, n(A)= 4
B= set of letters in the word ALGEBRA
B= {A,L,G,E,B,R} Therefore, n(B)= 6

12. DISJOINT SETS
• Two sets A and B are said to be disjoint. If they do not have any element in commom
• FOR EXAMPLE:
A= {x : x is a prime number }
B= {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.

13. POWER SET
• The collection of all subsets of set A is called the power set of A. It is denoted by p(A).
In P(A), every element is a set.
• FOR 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)]= 4= 22
In general, n[P(A)]= 2m where m is the number of elements in set A.

14. 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 ∪ or 𝜉
• FOR EXAMPLE
1. 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]
2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal
set is a set of all integers
3. If A= {a,b,c} B= {d,e} C={f,g,h,i}
Then U= {a,b,c,d,e,f,g,h,i} can be taken as universal set.