$$ X(Y+Z) = (XY) + (XZ) $$
I can’t seem to derive the proper steps to prove this equation using Boolean axioms. The hint I’ve been given is using demorgans laws proofs but I still can’t seem to figure it out. These are the axioms I’ve been given to prove this. I know that this is typically a given axiom but I have to use the others to prove that this is true.
$\endgroup$ 91 Answer
$\begingroup$I'm using the notation you use in the question.
It might not be obvious which law is each, since in the image your axioms have a different notation, but I thing with the tags in each step you'll get it.
\begin{align}
XY + XZ
&= (XY + X)(XY + Z) \tag{distributivity}\\
&= X (XY + Z) \tag{absorption}\\
&= X (Z + XY) \tag{commutativity}\\
&= X ((Z + X)(Z + Y)) \tag{distributivity}\\
&= (X(Z + X))(Z + Y) \tag{associativity}\\
&= X(Z + Y) \tag{absorption}\\
&= X(Y + Z). \tag{commutativity}
\end{align}
Notice you still have to pick the right law (there is a pair of each), except for the distributivity, in which case we're using here the one there we're not proving.
