Question

What is Idempotent law?

Answer 1

Idempotent can be said to be an operation which can be repeated again without changing the result. That is, the expression $x\circ x=x$ where $x$ comes in the domain of $\circ$ and comes as a reference for the law. So, regardless of the number of the operations done, the expression will always yield the same result.

Complete step-by-step solution:
According to the given question, we have to explain the idempotent law.
The term idempotent would mean the repetition of a single work and still getting the same result. The law is useful both in mathematics as well as in computers for Boolean algebra.
Idempotent law can be expressed as $a*a=a$, where $*$ can be any operation.
In mathematics, we use idempotent law in laws of sets, let’s take a look how it is used in the union of sets as well as the intersection of sets.

1. Union set –
   In union sets, we take all the elements present in both the sets involved in union operation.
   Let A be a set having elements 1, 2 and 3, that is, $A=\\{1,2,3\\}$
   If we take the union of A with A, we will get the same set as the elements are the same.
   So, we have,
   $A\cup A=A$
2. Intersection set -
   In intersection sets, we take only the common elements present in both the sets involved in union operation.
   Let A be a set having elements 1, 2 and 3, that is, $A=\\{1,2,3\\}$
   If we take the intersection of set A with set A, we will get the same set as the elements are the same.
   So, we have,
   $A\cap A=A$

Note: The only idempotent element number are 1 and 0 in the multiplication operation as they give 1 and 0 respectively when multiplied by itself, that is,
$1\times 1=1$
$0\times 0=0$
And no other number holds the idempotent law true.