The below given is the Idempotent Law in boolean algebra tutorial that provides proof for an Idempotent law. Also, the brief definition the law is provided for your reference. Feel free to refer this tutorial to get an overall idea of about the law.
In Boolean algebra, Idempotent Law states that combining a quantity with itself either by logical addition or logical multiplication will result in a logical sum or product that is the equivalent of the quantity .
A + A = A
A × A = A
Show that a + a = a in a boolean algebra.
Proof: We can consider 'a' in the RHS to prove the law. We can write, 'a' as, a + 0. So, a = a + 0 = a + (a. a' ) (According to first law of Complement, X • X' = 0)
= (a + a ). (a + a ' )
= (a + a ). 1 (According to the second law of Compelement, X + X' = 1)
= (a + a )
Since, from RHS a = a + a, therefore a + a = a
Note: The dual a . a = a