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Boolean behavior of the parallel input diode network at the transistor base: $$V_b = A + B$$
Operation of the common-emitter transistor stage configured as an inverter: $$Y = \overline{V_b}$$
Combining the two stages: $$Y = \overline{A + B}$$
Applying De Morgan's laws: $$Y = \bar{A} \cdot \bar{B}$$
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