A truth table proves what a circuit does for every possible input. With three inputs there are eight rows, and one slip in one row loses a mark. Learn a reliable method: count in binary, add a working column for each gate, then fill in the output.
A truth table shows the output for every combination of inputs. With n inputs there are 2n rows: 4 rows for two inputs, 8 rows for three. List the inputs in binary counting order (000, 001 ... 111) so no row is missed. Then add a working column for the output of each gate, working from the inputs towards the output, and use those columns to find the final output X.
Fig. 1: the truth table for X = (A AND B) OR NOT C, with a working column for each gate
Section 01
Count in binary.
Each row of inputs is a binary number. Writing the rows in counting order is the quickest way to be sure all eight are there and none is repeated.
Row
A
B
C
0
0
0
0
1
0
0
1
2
0
1
0
3
0
1
1
4
1
0
0
5
1
0
1
6
1
1
0
7
1
1
1
Column A: four 0s then four 1s. Column B: two 0s, two 1s, repeated. Column C: 0, 1 alternating.
Exam tables usually give the input columns already filled in this order: you complete the output column.
Examiner trap
Check the input columns before you start. If you write your own and repeat a row (or miss one), every output after it can be marked wrong.
Section 02
Working columns.
In the lab, pick a circuit and switch on the working columns. Fill in the output of each gate for all eight rows before you try the final output, then check your answers.
A
B
C
A XOR B
B OR C
X
0
0
0
0
0
0
0
0
1
0
1
0
0
1
0
1
1
1
0
1
1
1
1
1
1
0
0
1
0
0
1
0
1
1
1
1
1
1
0
0
1
0
1
1
1
0
1
0
Worked example Two working columns
Complete the truth table for X = (A XOR B) AND (B OR C).
Working column A XOR B: 1 only where A and B differ, rows 2 to 5: 0, 0, 1, 1, 1, 1, 0, 0.
Working column B OR C: 0 only where B = 0 and C = 0, rows 0 and 4: 0, 1, 1, 1, 0, 1, 1, 1.
X = 1 only where both working columns are 1: rows 2, 3 and 5.
Check row 4 (A = 1, B = 0, C = 0): A XOR B = 1 but B OR C = 0, so X = 0.
Answer: X = 0, 0, 1, 1, 0, 1, 0, 0
Your turn
Complete the output column for X = (NOT (A OR B)) AND C.
NOT (A OR B) is 1 only for rows 0 and 1 (A = 0, B = 0); AND C keeps row 1 only: 0, 1, 0, 0, 0, 0, 0, 0
Stage 1 · Learn
Check what the sim just showed you
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
How many rows does a truth table for a circuit with three inputs need?
Quick check+10 XP
Inputs are listed in binary counting order. What are A, B and C in the sixth row (row 5, counting from row 0)?
Quick check+10 XP
In the lab, what is a working column for?
Quick check+10 XP
For X = (A AND B) OR NOT C, what is X when A = 0, B = 1 and C = 0?
Section 03
From a circuit or a statement.
With a circuit, label the output of every gate with a letter or its expression, and make each label a working column. With a statement, write the logic expression first. The lab's problem statements reveal the truth table as the last step.
G
P
Q
P OR Q
M
0
0
0
0
0
0
0
1
1
0
0
1
0
1
0
0
1
1
1
0
1
0
0
0
0
1
0
1
1
1
1
1
0
1
1
1
1
1
1
1
Worked example From a statement
A machine M starts (M = 1) when the safety guard is closed (G = 1) and either start button P or start button Q is pressed (= 1). Complete the truth table.
Write the expression: M = G AND (P OR Q).
Working column P OR Q: 0 only where P = 0 and Q = 0: 0, 1, 1, 1, 0, 1, 1, 1.
M = 1 where G = 1 and P OR Q = 1: the last three rows.
Answer: M = 0, 0, 0, 0, 0, 1, 1, 1
Your turn
Using Fig. 1, give X for A = 1, B = 0, C = 0.
A AND B = 0, NOT C = 1, so X = 0 OR 1 = 1
Examiner trap
Read the meaning of 0 and 1 for each input. "The door is closed (D = 0)" means the condition is NOT D, not D.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 1 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
What is the output column of X = (A OR B) AND NOT C, for rows 000 to 111 in order?
Exam style+20 XP
For X = (A NAND B) XOR C, what is X when A = 1, B = 1 and C = 1?
Exam style+20 XP
In the truth table for X = (A AND B) OR (B AND C), how many rows have X = 1?
Exam style+20 XP
A drink is released (R = 1) when a coin is accepted (C = 1) and the drink is in stock (S = 1) and the machine is not faulty (F = 0). For which inputs is R = 1?
Skill unlocked
Completing truth tables, mastered.
This skill is now recorded on your profile. Keep the chain going.