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From Circuit Diagram to Boolean Expression

Label each gate output, write its expression from its inputs, and substitute until only the circuit inputs are left.

The Step-by-Step Method

To write the Boolean expression for a circuit: 1 Label the output of every gate (P, R, S \dots and X for the final output). 2 For each gate, write its expression using its own inputs. Start with the gates nearest the inputs. 3 Substitute each label into the next gate's expression. 4 Keep going until the final output is written only in terms of the inputs (A, B, C \dots). Use brackets whenever an OR expression goes into an AND gate — just like in normal maths, where 2 \times (3 + 4) is different from 2 \times 3 + 4 .

X = 0

Try it:

Write the expression for P, R, S and X on paper first. Then press the button to check.

Circuit 2 — four gates with every middle wire labelled.

X = 1

Try it:

Find the input combinations that make X = 0.

Circuit 3 — an XOR gate feeding a NAND gate.

Practice

An OR gate takes A and B. Its output P and input C go into an AND gate. What is X?

The Step-by-Step Method

To write the Boolean expression for a circuit: 1 Label the output of every gate (P, R, S \dots and X for the final output). 2 For each gate, write its expression using its own inputs. Start with the gates nearest the inputs. 3 Substitute each label into the next gate's expression. 4 Keep going until the final output is written only in terms of the inputs (A, B, C \dots). Use brackets whenever an OR expression goes into an AND gate — just like in normal maths, where 2 \times (3 + 4) is different from 2 \times 3 + 4 .

X = 0

Try it:

Write the expression for P, R, S and X on paper first. Then press the button to check.

Circuit 2 — four gates with every middle wire labelled.

A Circuit with XOR and NAND

X = 1

Try it:

Find the input combinations that make X = 0.

Circuit 3 — an XOR gate feeding a NAND gate.

Practice

An OR gate takes A and B. Its output P and input C go into an AND gate. What is X?