Suppose only one multiplexer and one inverter are allowed to be used to implement any Boolean function of $$n$$ variables. What is the minimum size of the multiplexer needed?

How many $$3$$ to $$8$$ decodes with an enable input are needed to construct to constant $$6$$ to $$64$$ line decoder without using any other logic gates

Let $$f\left( {w,x,y,z} \right) = \sum {\left( {0,4,5,7,8,9,13,15} \right).} $$ Which of the following expressions are NOT equivalent to $$f?$$
$$(P)\,\,\,$$ $$x'y'z' + w'xy' + wy'z + xz$$
$$(Q)\,\,\,$$ $$w'y'z' + wx'y' + xz$$
$$(R)\,\,\,$$ $$w'y'z' + wx'y' + xyz + xy'z$$
$$(S)\,\,\,$$ $$x'y'z' + wx'y' + w'y$$

Consider the following Boolean function with four variables
$$F\left( {w,\,x,\,y,\,z} \right) = \sum {\left( {1,\,3,\,4,\,6,\,9,\,11,\,12,\,14} \right)} $$ the function is