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🔗 Topic 10 Boolean logic · Page 10.8–10.10

10.8–10.10 Composition Lab

Join gates into circuits. Move between a logic expression, a truth table and a logic circuit — a skill that comes up regularly in Paper 2 logic questions.

⚠️ This lab builds on the Gates Lab. It assumes you know all six gate symbols and their truth tables. If that isn't solid yet, brush up in the Gates Lab first — or carry on here. Your call.
🧪 Exam Mode ON — recall from memory, then reveal
§1 · Topic Overview

Where this lab sits

In the Gates Lab you learnt the six gates one at a time. In this lab you join them into logic circuits with up to three inputs, and move between three ways of describing the same circuit:

  • Logic expression — e.g. X = (A AND B) OR NOT C
  • Truth table — every input combination and its output
  • Logic circuit — the gates drawn and wired together

The syllabus asks you to create a circuit, complete a truth table and write a logic expression from any of the others. These conversions come up regularly in Paper 2 logic questions.

🔗 The Circuit idea

"The gates compose. The circuit computes the function." All three forms describe the same circuit. Learn to move between them carefully and you have the core of this topic.

§2 · Learning Objectives

By the end of this lab you will be able to:

  • Read a logic expression and draw the circuit for it — as given, without simplifying
  • Read a logic circuit and write its logic expression in words
  • Complete a truth table from a logic expression or a circuit, using working-space columns
  • Write a logic expression from a truth table, using sum-of-products as a reliable method
  • Use brackets correctly — and avoid the B OR NOT C vs NOT (B OR C) trap
  • Work with circuits that have up to three inputs and one output
§3 · Key Terminology

Six new terms (on top of the Gates Lab terms)

Logic circuit — a combination of several logic gates that are linked together.
Intermediate output — the output of a gate inside a circuit that becomes the input of the next gate. Often labelled D, E, F.
Working space — truth-table columns for intermediate outputs. Some exam truth tables include them to help you track values.
Sum-of-products — a method for writing an expression from a truth table: for each row where the output is 1, write an AND of the inputs (a 1 as the letter, a 0 as NOT of the letter), then join those terms with OR.
Brackets — show which part of an expression is worked out first, and which gate each part feeds. Innermost brackets first, as in maths.
Final gate — the last gate in a circuit; its output is the circuit's output. Start here when writing an expression or drawing a circuit.
§4 · Core Theory · four skills

§4.1 · Circuit or expression → truth table

  1. Count the inputs. 2 inputs → 4 rows. 3 inputs → 8 rows. Set up the table first.
  2. Fill the inputs in binary-count order. 000 · 001 · 010 · 011 · 100 · 101 · 110 · 111. Same order every time.
  3. Label intermediate outputs. Where a gate's output feeds another gate, label it D, E, F and give it a working-space column.
  4. Fill each working column, one gate at a time. All eight rows before you move on.
  5. Fill the output column, using the working columns as the inputs to the final gate.

§4.2 · Circuit → expression

  1. Start with the output label and an equals sign: X = (leave it out if the question already gives it).
  2. Find the final gate — the one whose output is the circuit output. Write its name.
  3. Look at what feeds each input of the final gate. Each one becomes a bracketed part of the expression.
  4. Repeat for every gate inside those brackets until you reach the inputs A, B, C.
Example: an OR gate on A and B and a NOT gate on C both feed an AND gate → X = (A OR B) AND NOT C.

§4.3 · Truth table → expression (sum-of-products)

Sum-of-products is a reliable method: it always gives a correct expression, using only NOT, AND and OR.

  1. Look only at the rows where the output is 1.
  2. For each of those rows, write an AND term: a 1 is written as the letter, a 0 as NOT of the letter.
  3. Join the terms with OR. Result: X = (term 1) OR (term 2) OR …
Example: a 2-input table with output 1 at (0, 1) and (1, 1) gives X = (NOT A AND B) OR (A AND B).

§4.4 · Expression → circuit

The reverse of §4.2. Draw from the right (final gate) and work left.

  1. Find the outermost gate — the one joining the outermost brackets. Draw it on the right of the page.
  2. Each bracketed part is an input to that gate. Find its own outermost gate and draw it feeding in.
  3. Repeat for each bracket inside.
  4. Inputs A, B, C become lines on the far left. Every gate except NOT has exactly two inputs.

✋ Draw it as given

The syllabus says circuits must be drawn for the statement given, without simplification. Even if you can spot a shorter circuit that does the same job, draw the one the expression describes — gate for gate.

§5 · Worked Examples · six across the four skills

Worked step by step

Each example: given → steps → result → why it works.

§6 · Common Misconceptions · 10 traps

Where students lose marks (and how not to)

Some of these are noted in Cambridge examiner reports; the rest come from the syllabus and good exam habits. Every one has a fix.

§7 · Exam Focus

Wording you may see in questions

Typical instructions in logic questions, and what they ask you to do.

"Draw a logic circuit for this logic expression" — expression → circuit, gate for gate
"Do not attempt to simplify" — draw or write it exactly as given; don't shorten it
"Each logic gate must have a maximum of two inputs" — NOT has one input; every other gate has two
"Complete the truth table" — inputs in binary-count order, then the output for each row
"Working space" — columns for intermediate outputs; use them
"Write a logic expression for this logic circuit" — circuit → expression, starting from the final gate
Expressions in words: X = (A OR B) AND NOT C — use the gate names and brackets
§8 · Quick Knowledge Check · 8 questions

0 / 8 revealed

Q1 · How many rows does a truth table for a 3-input circuit have?
8. 2 inputs → 4 rows; 3 inputs → 8 rows (000 to 111).
Q2 · What is the first step when writing a logic expression from a circuit?
Find the final gate — the one whose output is the circuit output — and write X = followed by that gate. Then work back through its inputs.
Q3 · When drawing a circuit from an expression, which gate do you draw first?
The final (outermost) gate. Draw it on the right of the page and work backwards.
Q4 · When writing a logic expression from a truth table, which rows do you use?
The rows where the output is 1. Each one becomes an AND term in the sum-of-products method.
Q5 · Which gates does the sum-of-products method use?
NOT, AND and OR. It's a reliable method — it always gives a correct expression.
Q6 · In a 3-input truth table, in what order are the input rows listed?
Binary-count order: 000, 001, 010, 011, 100, 101, 110, 111 — A on the left, C on the right.
Q7 · What is an intermediate output?
The output of a gate inside a circuit that becomes the input of the next gate. Often labelled D, E, F.
Q8 · Should you simplify an expression before drawing its circuit?
No. Draw the circuit for the expression as given — the syllabus says without simplification.
§9 · Ready for the next lab?

✅ You can move between the three forms.

Expression ↔ truth table ↔ circuit: you use working columns, you know sum-of-products, you start from the final gate and you draw circuits as given.

Next up: the Problem Statements Lab — where you turn real-world rules into expressions, truth tables and circuits.

Previous · Gates Lab

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Topic 10 Path

10.1–10.7 Gates Lab
📍 10.8–10.10 Composition Lab (this page)
10.11 Problem Statements Lab
Next: Topic 11 Programming Scenarios

Cambridge syllabus

Covers Cambridge 0478 Topic 10, outcome 3: (a) create logic circuits, (b) complete truth tables and (c) write logic expressions — each from a logic expression, a logic circuit or a truth table. Circuits have up to three inputs and one output, and are drawn as given, without simplification. Problem statements are in the next lab.

🎓 Moving between the three forms

Learn slows down what Book Notes covered. When you're comfortable, head to Activities and open the Circuit Builder.

The three forms — same circuit, three views

FormExampleGood for
ExpressionX = (A AND B) OR NOT CWriting it down compactly
Truth table8 rows · A, B, C and X columnsChecking every case
Circuit3 gates: AND on A, B; NOT on C; both into an ORSeeing how it is built

The circuit for X = (A AND B) OR NOT C.

Draw it as given — don't simplify

🔍 Contrast box · as given vs simplified

Question: "Draw a logic circuit for X = (A AND B) OR (A AND NOT B)."

✅ As given

Four gates: an AND on A and B, a NOT on B, an AND on A and NOT B, and an OR joining the two ANDs. This is what the question asks for.

❌ Simplified

The output happens to equal A, so a student draws a single wire from A to X. Same truth table — but it isn't the circuit for the statement given, so it doesn't earn the marks.

The rule: one gate for each gate name in the expression, wired the way the brackets say.

Working columns — your scaffold

Any time a gate's output feeds another gate, give it a label — D, E, F — and a column in the truth table. Fill one column at a time, all rows, before moving on.

Example. For X = (A OR B) AND NOT C:

ABCD
(A OR B)		E
(NOT C)		X
(D AND E)
000010
001000
010111
011100
100111
101100
110111
111100

Notice. Column D only uses A and B. Column E only uses C. Column X only uses D and E. You never juggle three inputs at once.

📎 Useful background — other notations

Some books write AND as A.B, OR as A + B and NOT as a bar over the letter. In this course write logic expressions in words, using the gate names: X = (A OR B) AND NOT C.

Ready for Activities?

The Circuit Builder has 5 modes. Start with Modes 4 and 5 (completing truth tables). Modes 1, 2 and 3 unlock once you have checked an answer in both.

🎯 Syllabus links

This lab covers Topic 10, outcome 3:

  • 3(a) — create a logic circuit from an expression or a truth table
  • 3(b) — complete a truth table from an expression or a circuit
  • 3(c) — write a logic expression from a circuit or a truth table

Not in this lab: problem statements — those are in the Problem Statements Lab.

🎮 Circuit Builder

Five modes, one for each way of moving between the three forms. In Mode 3 your expression is checked for the same circuit: C OR (A AND B) matches (A AND B) OR C, but a simplified version that only behaves the same does not.

Unlocking: Modes 1, 2 and 3 unlock after you have checked an answer in Mode 4 and in Mode 5.

60-second Sprint

Quick single-row questions from circuits and expressions: work out X for one row. Streaks show at 3 · 6 · 10 correct. Sprint is for speed — it doesn't change your Mastery.

60Seconds left
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0Streak

🔗 Mode coverage

Answers from the Circuit Builder, Practice and Exam, by skill.

💡 Mode guide

Mode 4 · Circuit → Truth table. See a circuit and complete the output column.

Mode 5 · Expression → Truth table. See an expression and complete the output column. Use working columns on paper.

Mode 1 · Expression → Circuit. Read an expression and choose the circuit that matches it, from three. Sketch it on paper first.

Mode 2 · Truth table → Circuit. Read a truth table and choose the circuit that produces it.

Mode 3 · Circuit → Expression. Read a circuit and write its expression in words.

✎ Practice — 18 multiple-choice questions

Skills you get wrong come back more often. Filter by difficulty or run the whole pool. Missed questions are saved to Mistakes.

🧪 Practice Exam Mode ON. No multiple-choice options shown — recall the answer, then reveal and self-mark.

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📋 Exam-style questions — 12 questions

Paper 2 style questions on logic circuits. Answer every part, then press Mark my answer — circuits, truth tables and logic expressions are checked automatically.

📋 Exam stats

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ℹ️ About these questions

Written in the style of Cambridge Paper 2, where Boolean logic is assessed. In the real exam you draw circuits — sketch each one on paper first, then use the "choose the circuit" parts here to check your drawing.

🔄 Review — 10 traps + 6 memory triggers

Read these before your exam. Then look at Mistakes to see the ones you make.

10 traps to know

🧠 Memory triggers

Six one-liners to recite before your exam.

🎯 Your weakest skills

💔 Your logged mistakes

🏆 Mastery — revision checklist

Tick each item as you master it. In the skill grid, tap a status chip to set it yourself. When all 12 are ticked, the Module Mastered banner unlocks.

🏆 Module Mastered — 10.8–10.10 Composition Lab complete!

Revision checklist

Skill grid

Mastered = 3+ attempts and ≥70% accuracy. Tap the status chip to override it.

➡️ Next up: Problem Statements Lab

Turn real-world rules into logic expressions, truth tables and circuits.

📊 Session summary

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Complete Practice and the Circuit Builder to unlock achievements.Achievement

Topic 10 Path

10.1–10.7 Gates Lab
📍 10.8–10.10 Composition Lab (this page)
10.11 Problem Statements Lab
Next: Topic 11 Programming Scenarios

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