Deriving an Expression from a Logic Circuit — Boolean Logic and Expressions | IGCSE Computer Science

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Writing Boolean expressions from logic diagrams and simplifying using laws appear regularly.

Given a circuit, the safest method is to label each gate's output with its Boolean sub-expression, working from the inputs through to the final output.

Method. (1) Identify each gate and the signals feeding it. (2) Working left to right, write the output of each gate as a Boolean sub-expression using its input labels. (3) Use that sub-expression as the input to any gate it feeds. (4) The final gate's output is the expression for Q.

Example — a circuit has three gates:

Gate 1: an OR gate with inputs A and B. Output label: P1.
Gate 2: a NOT gate with input C. Output label: P2.
Gate 3: an AND gate with inputs P1 and P2. Output Q.

Writing each intermediate:

P1 = A + B
P2 = ¬C
Q = P1 · P2 = (A + B) · ¬C

Notice the brackets around (A + B). Without them the expression would read A + B · ¬C, which (by AND-before-OR precedence) means A + (B · ¬C). That is a completely different circuit.

Example — a three-gate circuit with mixed gates:

Gate 1: NAND of A and B. Output P1 = ¬(A · B).
Gate 2: NOR of B and C. Output P2 = ¬(B + C).
Gate 3: OR of P1 and P2. Output Q = ¬(A · B) + ¬(B + C).