Boolean Algebra Proof for a + a = a and (a * b)' = a' + b'

Question

Prove, for any element $a$ in a boolean algebra expression, that $a + a = a$. Prove also, for any two elements, $a$ and $b$, of a boolean algebra expression, that $(a * b)' = a' + b'$.

Answer 1

Idempotent law a + a = a

Proof: x + x

= (x + x) • 1

= (x + x) • (x + x')

= x + (x • x')

= x + 0 = x

And for other prove see de-morgan's law.

Answer 2

from right side

=a

=a+0

=a+(a.a')

=(a+a).(a+a')

=(a+a).1

=a+a