Q.In the combination of the following gates, write the Boolean equation for output Y in terms of input A, B, C. [figure: logic circuit — input A passes through a NOT gate; its output and input B feed a 2-input NOR gate (OR-shape with an output bubble); the NOR gate's output and input C feed a 2-input NAND gate (AND-shape with an output bubble) whose output is Y]

Boolean algebra (George Boole, 1854) is an algebra built entirely on the binary choice 0/1, governing the three basic operations NOT ($\overline{A}$), OR ($A+B$) and AND ($A\cdot B$). Its laws: COMPLEMENT law ($A\cdot\overline{A}=0$, $A+\overline{A}=1$); OR laws ($A+0=A$, $A+1=1$, $A+A=A$); AND laws ($A\cdot0=0$, $A\cdot1=A$, $A\cdot A=A$); COMMUTATIVE ($A+B=B+A$, $A\cdot B=B\cdot A$); ASSOCIATIVE ($A+(B+C)=(A+B)+C$, $A\cdot(B\cdot C)=(A\cdot B)\cdot C$); and DISTRIBUTIVE ($A(B+C)=AB+AC$, and the Boolean-specific $A+BC=(A+B)(A+C)$, which has no ordinary-arithmetic analogue). These laws simplify complicated logic expressions -- and hence the physical gate count needed to build them -- e.g. $AC+ABC=AC(1+B)=AC\cdot1=AC$ (using OR law 2 then AND law 2), collapsing a 3-input-AND-plus-2-input-AND-plus-OR circuit down to a single 2-input AND gate. …