Question

What is Idempotent law?

Answer 1

Hint: 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.