Theory:

Set operation is an operation used to construct new sets from given sets.
There are several fundamental operations for constructing new sets from given sets. They are:
  • Universal set.
  • Union of two sets.
  • Intersection of two sets.
  • Difference of two sets.
  • Symmetric difference of sets.
  • Complement sets.
  • Disjoint sets.
  • Overlapping sets.  
Universal set:
A universal set is the set of all elements under consideration, denoted by capital \(U\). All other sets are subsets of the universal set.
 
U= {1, 2, 3, 4, 5, 6, 7, 8, 9,10,12},A = {2,4,6,8},B = {10,12}

In the Venn diagram below, represents a universal set.
 
Xcvv.svg
Union of two sets:
Let \(A\) and \(B\) be any two sets. The union of \(A\) and \(B\) is the set consists of all elements of \(A\) and \(B\); common elements being taken only once. The symbol \(∪\) is used to denote the union. Symbolically, we write \(A ∪ B\) and usually read as \(A\) union \(B\).
 
Symbolically, we write \(A\cup B = \{x: x\in A\ \mathrm{or}\ x\in B\}\)
 
Let \(A=\{2,\ 4,\ 6,\ 8\}\) and \(B=\{6,\ 8,\ 10,\ 12\}\).
 
Then, \(A\cup B=\{2,\ 4,\ 6,\ 8,\ 10,\ 12\}\)
 
Note: Common elements \(6\) and \(8\) have been taken only once while writing \(A ∪ B\).
 
In the Venn diagram below \(A\) and \(B\) are represented.
 
Xcvxdv_3.png  
  
In the Venn diagram below, the area in the blue colour represents \(A ∪ B\).
 
Images.png
Intersection of two sets:
The intersection of sets \(A\) and \(B\) is the set of elements which are common in \(A\) and \(B\). The symbol \(∩\) is used to denote intersection.
 
Symbolically, we write \(A\cap B=\{x:x\in A\ \mathrm{and}\ x\in B\}\).
 
Numerical: Let \(A=\{2,\ 4,\ 6,\ 8\}\) and \(B=\{6,\ 8,\ 10,\ 12\}\).
 
Then, \(A\cap B=\{6,\ 8\}\). 
 
In the Venn diagram below \(A\) and \(B\) are represented.
 
Xcvxdv_3.png
 
In the Venn diagram below, the area in the blue colour represents \(A ∩ B\).
 
Zx (1).png
 
Important!
The relation "is an element of", also called set membership, is denoted by the symbol "\(∈\)", and the relation "is not an element of", is denoted by the symbol "\(∉\)".