1. Introduction to set operations I

There are several fundamental operations for constructing new sets from given sets. They are:

A universal set is the set of all elements under consideration, denoted by capital \(U\). All other sets are subsets of the universal set.

 

In the Venn diagram below, represents a universal set.

 

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, the area in the blue colour represents \(A ∪ B\).

 

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, the area in the blue colour represents \(A ∩ B\).