Rewriting a boolean expression in SOP form

Question

I have started a discrete math module for my computer science course and I'm having a little trouble using the identity, idempotent and complement laws to convert a boolean expression into sum of products (SOP) form.

$F(x,y,z) = (x+y')*z$
$=(z*x)+(z*y')$
$=(z+z)*(z+y')*(z+x)*(x+y')$

This is where I'm stuck. I could simplify it but I am not sure how to get to $(x'y'z)+(xy'z)+(xyz)$ that i got with a truth table.

Any guidance would be greatly appreciated.

Kind regards, Luke

Answer 1

first find $x+y'$ in SOP form.
$x+y'=x(y+y')+y'(x+x')$
$=xy+xy'+xy'+x'y'$
$=xy+xy'+x'y'$

multiply both sides by $z$.
$(x+y')z=(xy+xy'+x'y')z$
$=xyz+xy'z+x'y'z$

Answer 2

After your first step you have $(z*x)+(z*y')$. That is in SOP, so you are done!