AN INTERESTING INTRODUCTION TO THE CONCEPT OF ‘SET THRORY’
Published in May2021 issue of ‘VEDIC MATH WORLD’ – magazine

INTRODUCTION
Everything in the world can be represented in the form of set and subset. Subset is a part of set. in understandable manner if we put this concept than man is subset of human beings and human being is subset of living beings.
A German mathematician named, Georg cantor(1845-1918), initiated the theory of set or ‘set theory’. When he was working on “problems on trigonometric series” he found sets, that become one of the most fundamental concepts in mathematics. collection of objects or group of objects is called set. the objects are termed as elements or member of sets.

TYPES OF SETS:
Equal set: Two sets are equal if they have same elements
Equivalent set: Two sets are equivalent if they have same number of elements
Empty set: It has no elements
Finite set: The number of elements is finite
Infinite set: The number of elements is infinite
Subset: When all the elements of set A belong to set B, then A is subset of B
Power set: A set of every possible subset.
Singleton set: It has one only element
Universal set: Any set that contains all the sets under consideration.

FORMULA:
n ( A ∪ B ) = n(A) + n(B) – n (A ∩ B)
n (A ∪ B) = n (A – B) + n (B − A) + n(A ∩ B)
n (U) = n (A) + n (B) – n (A ∩ B) + n (( A ∪ B ) c )
n (A) = n( A – B) + n ( A ∩ B )
n (B) = n ( B – A) + n ( A ∩ B )
n ( A ∪ B ) = n ( A ) + n ( B )   {when A and B are disjoint sets}
n ( A – B ) = n ( A ∩ B ) – n ( B )
n ( A – B ) = n ( A ) – n ( A ∩ B )
n ( Ac ) = n ( U )− n ( A )
If A ∩ B = ∅, then n ( A ∪ B ) = n(A) + n(B)
n ( A ∪ B ∪ C ) = n(A) + n(B) + n(C) – n ( A ∩ B) – n ( B ∩ C) – n ( C ∩ A) +  n ( A ∩ B  ∩ C)

SET OPERATIONS
Union of sets
Intersection of sets
Complement of sets
Difference of sets

UNION OF SETS
There are set A and set B are given. The set which contains all the element of set A and set B is called A union set. which is denoted as  A ∪ B.
Example: Set A = {7,8,9} and B = {4,5,6}, then A union B is:
A ∪ B = {4,5,6,7,8,9}

INTERSECTION OF SETS
There are two set A and set B are given. The set which contains only the common element between set and set B is called as A intersection B. it is denoted as A ∩ B.
Example: Set A = {7,8,9} and B = {4,5,6}, then A intersection B is:
           A ∩ B = { } or Ø
Since A and B do not have any elements in common, so their intersection will give null set.

QUESTION_1
In a school there are 200 students and every student study at least one subject. 60 students study physics, and 80 students study biology. How many students study both physics and biology?
SOLUTION:
n ( A ∪ B ) = n(A) +n(B) – n (A ∩ B)
100 = 60 + 80 – n (A ∩ B)
n (A ∩ B) = 140 -100 = 40

COMPLEMENT OF A SET DEFINITION
There are U a universal set and A as the subset of universal set given. So, the members of universal set which are not part of A is called as complement of A.
A′ = {x : x ∈ U and x ∉ A}
As an Alternative it can be said that the difference of the universal set U and the subset A gives us the complement of set A.
Example:
U of all-natural numbers less than or equal to 20. Let the set A which is a subset of U be defined as the set which consists of all the prime numbers.
U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}
Thus, we can see that A = {2,3,5,7,11,13,17,19}
Now the complement of this set A consists of all those elements which is present in the universal set but not in A.
so, A′ = {1,4,6,8,9,10,12,14,15,16,18,20}

QUESTION_2
U is the universal set which consists of all the integers greater than 2 but less than or equal to 10. Let A and B be the subsets of U defined as:
A = {x : x ∈ U and x is a perfect square}
B = 3,5,7,9,
Find the complement of sets A and B and the intersection of both the complemented sets.
SOLUTION:
The universal set is defined as:
U={3,4,5,6,7,8,9,10}
A = {4,9} and
B = {6,7,8}
The complement of set A(A′ ) is defined as:
A′ = {x : x ∈ U and x ∉ A}
Therefore, A′ = {3,5,6,7,8,10}
Similarly, the complement of set B can be given by:
B′ = {3,4,5,9,10}
The intersection of both the complemented sets is given by A′∩B′.
A’ ∩ B’= {3,5,10}
As from the question we can see that set A is a subset of the set U than the set A′ is also the subset of U.

DIFFERENCE OF SETS
There are two sets, set A and set B are given. The elements which are present in set a but not in set B is called difference of two set A and B.
The elements which are given in set B but not in set A are called difference of set B and set A.

QUESTION_3:
Let A = {3 , 4 , 5 , 8 , 9 , 11 , 12 } and B = {1 , 2 , 3 , 4 , 5 }. Find out A – B and B – A.
SOLUTION:
A – B = {8 , 9 , 11 , 12} as these elements belong to A but not to B
B – A ={1,2} as these elements belong to B but not to A.