Learning Objectives
- Translate a written problem into variables, an expression, a truth table and a circuit.
- Choose gates from precise wording such as both, at least one, neither and exactly one.
- Check a complete solution against all possible input combinations.
- Apply an efficient method to Cambridge-style Boolean logic questions.
Key Terms
- Problem statement
- A written description of the conditions that control an output.
- Requirement
- A rule specifying when the output must be 0 or 1.
- Translation
- Conversion of words into variables and gate operations.
- Validation of a circuit
- Checking that the circuit produces the required output for every input combination.
- Integrated question
- A task linking a statement, expression, circuit and truth table.
- Without simplification
- Drawing the direct structure described rather than reducing it with Boolean algebra.
The Complete Five-Step Method
Step 1: define each input and output, including exactly when its value is 1. Step 2: split the problem statement into smaller logical clauses. Step 3: translate each clause into a gate operation and write a bracketed expression. Step 4: draw the circuit from the expression without simplification. Step 5: construct or complete the truth table and verify the required behaviour.
The same method works in reverse. If the question begins with a circuit, trace it to form a truth table and expression. If it begins with a truth table, identify a gate pattern or construct direct output-1 conditions.
A systematic method is more reliable than trying to guess the final output.
Logical Wording Guide
| Wording | Likely Operation |
|---|---|
| not A | NOT A |
| A and B; both A and B | AND |
| A or B; at least one; one or both | OR |
| not both; false only when both true | NAND |
| neither A nor B; true only when both false | NOR |
| either A or B but not both; exactly one | XOR |
Example Security Statement
A warning siren X sounds when the system is armed A and either the door D is open or the window W is open. Definitions: A = 1 when armed, D = 1 when door open, W = 1 when window open.
The inner alternative is D OR W. The system must also be armed, so X = A AND (D OR W).
Draw OR for D and W, then AND its result with A. This uses two gates and three inputs. The truth table should output 1 only in rows where A = 1 and at least one of D or W = 1.
Security Truth Table
| A | D | W | P = D OR W | X = A AND P |
|---|---|---|---|---|
| 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 |
Example With Inversion And XOR
A status lamp turns on when exactly one of sensors A and B is active and the disable input C is not active. The expression is X = (A XOR B) AND (NOT C).
Draw XOR for A and B, NOT for C, and AND as the final gate. The phrase exactly one indicates XOR, while not active indicates NOT.
Check the key rows: if A and B are both 1, XOR is 0 and the lamp is off. If exactly one is 1 but C = 1, NOT C is 0 and the lamp is off.
Working From A Given Truth Table
First check whether the output column matches a single known gate. If not, list every row where output equals 1. Write a direct condition for each row, using NOT for input 0 and the variable for input 1. Join within a row by AND and between rows by OR.
The resulting circuit may contain several gates. Draw it directly and do not simplify. Ensure that only two-input gates are used.
A truth table is the final authority for behaviour. Any proposed expression or circuit must produce every listed output.
Examination Presentation
Label gate symbols clearly and draw them at a size where bubbles and the extra XOR curve are visible. Label input wires and the final output.
In truth tables, use only 0 and 1. Align rows and show intermediate headings such as P = A AND B. Where a question gives a partially completed table, keep the supplied input order.
In written definitions, state the complete condition. For AND, write output 1 only when both inputs are 1. For OR, write output 1 when at least one input is 1. Avoid vague phrases such as it joins inputs.
Checking Before Finishing
Check 1: Are all required input combinations included? Check 2: Is every NOT or output bubble applied at the correct stage? Check 3: Are non-NOT gates limited to two inputs? Check 4: Does the circuit have one final output? Check 5: Has the given structure been preserved without simplification?
Finally, test one row that should output 1 and one that should output 0 according to the original statement. For a three-input problem, check edge cases such as 000 and 111 because they often expose OR/XOR or NAND/NOR confusion.
Scope Of Topic 10
The required topic consists of identifying and using standard gate symbols; defining NOT, AND, OR, NAND, NOR and XOR; creating circuits from statements, expressions and truth tables; completing truth tables from those sources; and writing expressions from statements, circuits and truth tables.
Circuits are limited to a maximum of three inputs and one output. NOT has one input; all other gates have two. Circuits should be drawn as stated without simplification.
Advanced Boolean algebra laws, Karnaugh maps, electronic transistor design and gate propagation timing are not required.
Worked Examples
Full Translation
Question: A cooling fan F operates when temperature T is high or when manual switch M is on, provided safety lock S is not active. Write the expression and describe the circuit.
- Combine the two alternative starting conditions: T OR M.
- Invert the safety lock: NOT S.
- Both the alternative condition and safe state are required, so use AND.
Answer: F = (T OR M) AND (NOT S). Draw OR for T and M, NOT for S, then feed both results into AND.
Choosing A Gate From A Statement
Question: An output is 1 only when neither sensor A nor sensor B is active. Which gate is required?
- Neither means A = 0 and B = 0.
- The output is 1 only for input 00.
- That is the NOR pattern.
Answer: NOR.
Examination Guidance
- Define the value 1 for every input before translating the statement.
- Underline words that indicate gate operations.
- Use brackets to preserve clause structure.
- Show intermediate truth table columns for each gate.
- Check the circuit against all rows and the original wording.
- Remember Topic 10 is assessed in Paper 2 with Topics 7 to 9.
Common Mistakes
- Choosing a gate from one word without reading the full statement.
- Failing to define whether an input is 1 for a safe or unsafe state.
- Using XOR for ordinary inclusive or.
- Simplifying the circuit or using out-of-syllabus Boolean algebra.
- Drawing more than three inputs or more than one output.
- Using three-input gate symbols.
Knowledge Check
1. What is the first step with a problem statement?
2. What gate is suggested by neither A nor B?
3. What gate is suggested by exactly one?
4. What are the circuit limits?
5. Name two topics outside the required scope.
