$($a$)$ Draw the circuit corresponding to the following boolean expression:

$P=(a+b)*(\frac{?}{\text{b}\text{a}\text{r}}(c*(\frac{?}{\text{b}\text{a}\text{r}}(c+b))))$

$($b$)$ For the circuit above, draw the truth table for the output of each gate.

$($c$)$ Show using Boolean logic properties $($truth table method will not be accepted$)$

that $P($the Boolean expression above$)$ is equivalent to the Boolean expression

$Q=a+b$

and draw the corresponding circuit.