Circuit Analysis using Boolean Algebra:

• The top NAND gate has both of its inputs tied together to input $A$. Its output is:
  $$Y_1 = \overline{A \cdot A} = \bar{A}$$
• The bottom NAND gate has both of its inputs tied together to input $B$. Its output is:
  $$Y_2 = \overline{B \cdot B} = \bar{B}$$
• These two inverted outputs are fed into the third NAND gate. The final output $Y$ is:
  $$Y = \overline{Y_1 \cdot Y_2} = \overline{\bar{A} \cdot \bar{B}}$$
• Applying De Morgan's Law ($\overline{X \cdot Y} = \bar{X} + \bar{Y}$):
  $$Y = \overline{\bar{A}} + \overline{\bar{B}} = A + B$$
  This is precisely the Boolean expression for an OR gate.

Circuit Analysis:

• Let node $B$ be at reference potential $V_B = 0\,\text{V}$.
• The battery maintains the top wire at $24\,\text{V}$.
• The junction between the two vertical $8\,\text{k}\Omega$ resistors is at a positive potential relative to $B$.
• Because the p-side (anode) of the diode is connected to this positive junction and the n-side (cathode) connects to node $A$ (which is connected to $0\,\text{V}$ via the right $8\,\text{k}\Omega$ resistor), the diode is FORWARD BIASED.
• Assuming an ideal diode, it acts as a short circuit with zero resistance.
  6. Consequently, the right branch $8\,\text{k}\Omega$ resistor is connected in parallel with the bottom $8\,\text{k}\Omega$ resistor across terminals $A$ and $B$:
  $$R_{\text{parallel}} = \frac{8\,\text{k}\Omega \times 8\,\text{k}\Omega}{8\,\text{k}\Omega + 8\,\text{k}\Omega} = 4\,\text{k}\Omega$$
  7. The total equivalent circuit resistance is:
  $$R_{\text{total}} = 8\,\text{k}\Omega + R_{\text{parallel}} = 8 + 4 = 12\,\text{k}\Omega$$
  8. Using the voltage divider rule, the potential across the parallel combination (between $A$ and $B$) is:
  $$V_{AB} = 24\,\text{V} \times \frac{4\,\text{k}\Omega}{12\,\text{k}\Omega} = 24 \times \frac{1}{3} = 8\,\text{V}$$

Matching semiconductor devices with their primary functions:

• Zener Diode: Designed to operate in reverse breakdown region as a Voltage Regulator (Matches II).
• LED (Light Emitting Diode): Emits infrared radiation when forward-biased, widely used in optical Remote Controls (Matches III).
• Rectifier (p-n junction diode arrangement): Converts alternating current (AC) to direct current (DC) (Matches IV).
• Photodiode: Operates in reverse bias to Detect optical signals / light intensity changes (Matches I).

Correct match: A-II, B-III, C-IV, D-I.