Solution:
(a) Input for the circuit: $A$ and $B$
Output for the circuit: $Y$
The left part of the circuit functions as a NOR gate, while the right half of the circuit functions as a NOT gate.
We can show it in the figure as:
Now, the output for the NOR gate will be $\overline{A+B}$
We are giving the output from NOR gate to the input at NOT gate. So, its output will be: $\overline{\overline{A+B}}=\mathrm{A}+\mathrm{B}$
$\therefore Y=A+B$
Hence, we proved this circuit functions as an OR Gate.
(b) Input for the circuit: $A$ and $B$
Output for the circuit: $Y$
The output from the first half of the circuits passes to the input of the second half, which acts as a NOR gate, as shown in the below figure:
Hence, the output of the given circuit can be written as:
$Y=\overline{\bar{A}+\bar{B}}=\overline{\bar{A}} \cdot \overline{\bar{B}}=A \cdot B$
Hence, we proved this circuit functions as an AND gate.