Why Does a' * a' = a'? Explained!

[magnifik]

why does a' * a' = a' ??

this isn't a homework problem, but i am just wondering why because i used this to simplify another boolean equation. oh, and what's the difference between a' * a' and (a')' ? how do you even get (a')'?? confused :\

 

[ystael]

Every element in a Boolean algebra is idempotent: $$xx = x$$ for every $$x$$.

$$a'a'$$ is the product of $$a'$$ with itself; $$(a')' = a$$ is $$a$$ complemented twice. Complement twice is the identity (complement is an involution).

 

[tnutty]

Make a truth table to see why.
X can equal only 1 or 0 thus the truth table for x*x is below :
--------
x x x*x
0 0 0
1 1 1
-------

As you see if X = 0, then X*X = 0 = X, and if X = 1, then X*X = 1 = X. Thus X*X = X