Boolean Algebra Distributive Property over More Than Two Literals?

Question

I am currently studying Boolean Algebra. While solving my practice questions, I came to wonder if I am properly using distributive law. So, distributive law states that
a. $·$ is distributive over $+$ : $x ( y + z ) = xy + xz$
b. $+$ is distributive over $·$ : $x + ( y · z ) = (x + y) · (x + z)$
(from http://www.ctp.bilkent.edu.tr/~yavuz/BOOLEEAN.html)

My solving process:

Is this correct as well?

Answer 1

To prove for three literals, just use associativity and the reasoning using two literals. \begin{align} \bar x+xy\bar z &= \bar x+x(y\bar z) = (\bar x + x) \cdot (\bar x + y\bar z) \\ &=(\bar x + x) \cdot \big((\bar x + y)\cdot(\bar x + \bar z)\big) \\ &=(\bar x + x) \cdot (\bar x + y) \cdot (\bar x + \bar z) \end{align} And then it follows your reasoning.

Answer 2

Yes, it's correct. But you can also convince yourself with the end result, handling $a:=y\bar z$ as one element: $$\bar x+xa=(\bar x+x)(\bar x+a) =\bar x+a $$