$$ X(Y+Z) = (XY) + (XZ) $$
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.
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.