Question

Which is true according to idempotent law?
A. $a + a = a$
B. $a + b = a$
C. $a + ab = a$
D. None of these

Answer 1

Hint: Use the fundamental definition of Idempotence in mathematics. The property of Idempotence of a certain operation that they can be applied to multiple times without affecting the result or initial application. An example of Idempotence can be given as $0 + 0 = 0$, $0 \times 0 = 0$ and $1 \times 1 = 1$.

Complete step-by-step solution:
Idempotence is the property of certain operations in mathematics that can be applied multiple times without changing the result beyond the initial application or number taken, the concept of Idempotence arises in several places in abstract algebra and fundamental programming where we don’t need to change the result. So, it can be defined mathematically as
$ \Rightarrow x \times x = x$
It means if all the elements are the same with respect to x, then x is called idempotent. Examples can be given as:
The natural number 1 is an idempotent element with respect to multiplication $1 \times 1 = 1$ and so is 0 $\left( {0 \times 0 = 0} \right)$, but no other natural number.
An identity element ‘e’, if it exists, is idempotent if \[e \times e = e\].
Now, coming to the options given in the problem. Here, we observed that option(a) i.e., \[a + a = a\] is showing property of Idempotent law as the sum of numbers a and a is a [0 can be an example]. So, it will be the correct option.
For further options i.e., \[a + b = a\] or \[a + ab = a\], we observe that the operation ‘ + ‘ between a and b or a and ab cannot be an idempotent as both the terms are not same i.e., a and b are not same in the first case and a and ab are not same in another case. So, these cannot show an idempotent property.

Option A is the correct answer.

Note: Don’t be confused by the ‘+’ sign between ‘a’ and ‘a’. It’s just acting as an operational tool. And one may think that how the sum of ‘a’ and ‘a’ can be equal to a, it should be equal to ‘2a’ which is wrong as ‘+’ is used here as a summation sign but we need to guess whether there is any number whose sum is equal to the same number (example: 0).