The simultaneous equations on the Boolean variables $$x, y, z$$ and $$w,$$ $$$x+y+z=1$$$ $$$xy=0$$$ $$$xz+w=1$$$ $$$xy + \overline z \overline w = 0$$$
have the following for $$x, y, z$$ and $$w,$$ respectively.

Which of the following sets of component(s) is/are sufficient to implement any arbitrary Boolean function?

Consider the logic circuit shown in Figure below. The functions $${f_1},$$ $${f_2}$$ and $$f$$ (in canonical sum of products form in decimal notation) are:

$${f_1}\left( {w,\,x,\,y,\,z} \right) = \sum {8,9,10} $$
$${f_2}\left( {w,\,x,\,y,\,z} \right) = \sum {7,8,12,13,18,15} $$
$$f\left( {w,\,x,\,y,\,z} \right) = \sum {\left( {8,9} \right)} $$

The function $${f_3}$$ is

Let $$f\left( {x,y,z} \right) = \overline x + \overline y x + xz$$ be a switching function. Which one of the following is valid?