Complete a truth table from a problem statement, logic expression or logic circuit

Boolean Logic – Cambridge IGCSE 0478

Learning Objective

Students will be able to complete a truth table when given a problem statement, a logical expression, or a logic circuit, and to move confidently between these three representations.

Key Concepts

• Boolean variables have only two possible values: True (1) or False (0).
• Only two‑input logic gates (except NOT) are used in the IGCSE exam. A circuit may contain a maximum of three inputs and one output.
• Standard logical operators required by the syllabus and their Cambridge symbols:

Operator	Cambridge Symbol	Gate	Expression
AND	∧ (or &)	AND	A ∧ B
OR	∨ (or +)	OR	A ∨ B
NOT	¬ (or overline)	NOT	¬A
XOR	⊕	XOR	A ⊕ B
NAND	↑	NAND	¬(A ∧ B)
NOR	↓	NOR	¬(A ∨ B)

Optional extensions (not required for the exam): Implication (A→B) and Equivalence (A↔B). These may appear in enrichment activities but are not part of the syllabus.

Gate‑Symbol Cheat‑Sheet

Gate	Graphic Symbol	Boolean Symbol
AND		∧ (or &)
OR		∨ (or +)
NOT		¬ (or overline)
XOR		⊕
NAND		↑
NOR		↓

Truth Tables for Individual Gates

Inputs	AND (∧)	OR (∨)	XOR (⊕)	NOT (¬)	NAND (↑)	NOR (↓)
A=0, B=0	0	0	0	¬A=1	1	1
A=0, B=1	0	1	1	¬B=0	1	0
A=1, B=0	0	1	1	¬A=0	1	0
A=1, B=1	1	1	0	¬B=0	0	0

Structure of a Truth Table

For n input variables there are 2ⁿ rows. Each row lists a unique combination of inputs and the resulting output.

Input A	Input B	Output F
0	0
0	1
1	0
1	1

From a Problem Statement → Expression → Circuit → Truth Table

1. Identify the variables. Give each a clear letter.
2. Write the logical expression. Use the symbols from the cheat‑sheet.
3. Draw the circuit. Replace each operator with its two‑input gate symbol (NOT is a single‑input gate).
4. Complete the truth table. List all input combinations, evaluate sub‑expressions step‑by‑step, and record the final output.

Worked Example 1 – Simple AND

Statement: “The alarm sounds if the door is open and the motion sensor detects movement.”

• Variables: D = door open, M = motion detected, A = alarm sounds
• Expression: A = D ∧ M
• Circuit: two inputs → AND gate → output A

D	M	A = D ∧ M
0	0	0
0	1	0
1	0	0
1	1	1

Worked Example 2 – XOR (“Door or Window, but not both”)

Statement: “The alarm sounds if either the door is open or the window is open, but not both.”

1. Variables: D = door open, W = window open, A = alarm sounds
2. Expression (two equivalent forms):
   • A = (D ∨ W) ∧ ¬(D ∧ W)
   • A = D ⊕ W (XOR)
3. Circuit: two inputs → XOR gate → output A

D	W	D ∨ W	D ∧ W	¬(D ∧ W)	A = (D ∨ W) ∧ ¬(D ∧ W)
0	0	0	0	1	0
0	1	1	0	1	1
1	0	1	0	1	1
1	1	1	1	0	0

Worked Example 3 – NAND

Statement: “The alarm sounds unless both the door and the window are open.”

• Variables: D = door open, W = window open, A = alarm sounds
• Expression: A = ¬(D ∧ W)  or  A = D ↑ W (NAND)
• Circuit: two inputs → NAND gate → output A

D	W	D ∧ W	A = ¬(D ∧ W)
0	0	0	1
0	1	0	1
1	0	0	1
1	1	1	0

Worked Example 4 – NOR

Statement: “The light stays off unless either switch S₁ or switch S₂ is turned on.”

• Variables: S₁, S₂ = switches, L = light on
• Expression: L = ¬(S₁ ∨ S₂)  or  L = S₁ ↓ S₂ (NOR)
• Circuit: two inputs → NOR gate → output L

S₁	S₂	S₁ ∨ S₂	L = ¬(S₁ ∨ S₂)
0	0	0	1
0	1	1	0
1	0	1	0
1	1	1	0

Completing a Truth Table from a Logical Expression

Break the expression into sub‑expressions, evaluate them column by column, and finally record the output.

Example Expression: F = (P ∨ Q) ∧ ¬R

P	Q	R	P ∨ Q	¬R	F
0	0	0	0	1	0
0	0	1	0	0	0
0	1	0	1	1	1
0	1	1	1	0	0
1	0	0	1	1	1
1	0	1	1	0	0
1	1	0	1	1	1
1	1	1	1	0	0

Completing a Truth Table from a Logic Circuit

1. Label every input on the circuit diagram.
2. Write the logical expression by replacing each gate with its symbol.
3. Use the “Expression → Table” method above.

Example Circuit (same expression as above):

Expression: F = (X ∨ Y) ∧ ¬Z → truth table identical to the previous example.

Deriving a Logical Expression from a Truth Table

Given a table, write the minimal Boolean expression in **sum‑of‑products** (SOP) form.

Mini‑Exercise

A	B	F (XOR)
0	0	0
0	1	1
1	0	1
1	1	0

Solution outline:

• Rows where F = 1: (0,1) and (1,0).
• Product terms:
  • (0,1): ¬A ∧ B
  • (1,0): A ∧ ¬B
• Combine with OR: F = (¬A ∧ B) ∨ (A ∧ ¬B), which simplifies to F = A ⊕ B.

Practice Questions

1. Complete the truth table for G = ¬(A ∧ B) ∨ (C ∧ ¬D).
2. A circuit has three inputs P, Q, R. P and Q go into an XOR gate; the result is ANDed with R. Write the logical expression and fill the truth table.
3. Problem statement: “A student passes if they score at least 50 % in **both** the written test and the practical test, **or** they achieve a perfect score in the written test.”
   • Define variables.
   • Write the expression.
   • Draw a simple circuit (max three inputs, two‑input gates only).
   • Complete the truth table.
4. Given the truth table below, write the corresponding Boolean expression (use sum‑of‑products).

   X	Y	Z	F
   0	0	0	0
   0	0	1	1
   0	1	0	1
   0	1	1	0
   1	0	0	1
   1	0	1	0
   1	1	0	0
   1	1	1	0

Summary Checklist – From Any Starting Point to a Completed Truth Table

• Identify **all** input variables (maximum three for the exam).
• Translate the English statement into a Boolean expression using the official Cambridge symbols.
• If you start from a circuit, label the inputs and write the equivalent expression.
• List every possible combination of inputs (2ⁿ rows).
• Evaluate sub‑expressions column by column – write intermediate results if it helps.
• Record the final output for each row.
• Cross‑check:
  • NOT columns are exact opposites of the original variable.
  • AND column is 1 only when **all** its inputs are 1.
  • OR column is 0 only when **all** its inputs are 0.
  • XOR column has an odd number of 1’s.
  • NAND and NOR are the opposites of AND and OR respectively.