
Operations on sets
Operations on sets can be done by combining two or more sets in different ways to produce another set. Let us discuss the basic operations here:
Union of sets:
For two sets A and B, the operation union is defined as,the set which are containing all the elements of set A and set B, without repeating elements.
The symbol for union is '∪'.
Symbolically we write A union B as A∪B and the definition is written in Roster form as,
A∪B={x:x∈A or x∈B}.
Examples:
1.Find the union of the sets
A={x:x∈N;5≤x≤10}
B={x:x=2n, 5≤x≤10, for x∈N}
Solution:
A={5,6,7,8,9,10}
B={6,8,10}
So,A∪B={5,6,7,8,9,10}
2. For the sets, P={a,e,o}and Q={i,o,u}, find P∪Q.
Solution:
P={a,e,o}
Q={i,o,u}
so,P∪Q = {a,e,i,o,u}
Note:
In both the examples, the elements are not repeated. That is the common elements are written only once.
Intersection of sets:
In the topic operations on sets next we are going to see intersection of two sets.For sets A and B, the operation intersection is defined as, the set which contains the common elements in both A and B.
The symbol to denote the intersection is '∩'.
Symbolically we write A intersection B as, A∩B and the definition for intersection is defined in Roster form as,
A∩B={x:x∈A, and x∈B} .
Examples:
Difference of sets:
For sets A and B,
Examples:
1.If set A={1,3,5} and B={5,6,7,8}, find AB and BA.
Solution:
A={1,3,5}
B={5,6,7,8}
5 is the common element in A and B
so,AB={1,3} and BA={6,7,8}.
 2. For the given sets P={a,e,i} and Q={b,c,d} find PQ and QP.
Solution:
P={a.e,i}
Q={b,c,d}
P and Q are disjoint sets, as there is no common element between them.
So, PQ={a,e,i}=P it self.
QP={b,c,d}=Q it self .
Complement of a set:
Let U be the universal set and A is a subset of U. Then the complement of set A, which is denoted as A'is set all elements which are not elements of A but elements in the set U.
Example:
1.If U={1,2,3,4,5,6,7,8,9,10} and A={2,3,4,8,9,10}. Find A'.
Solution:
U={1,2,3,4,5,6,7,8,9,10}
A={2,3,4,8,9,10}
For A' we have to write the elements of the set U which are not elements of A.
So, A'={1,5,6,7}
2. Find the complement of set B=set of all even natural numbers where, U= set of all natural numbers.
Solution:
U=N
A={2,4,6,8,....}
So, A'={1,3,5,7,9,..}=Set of all odd natural numbers.
Note:
 Complement of a universal set is the empty set.
 A set and its complement are always disjoint sets.
These are the important topics in operations on sets
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