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J277 2.4.1 Combined Boolean operators and truth tables
Part 2 of 4 · J277 2.4 · Boolean logic
Expressions that use more than one operator, worked with intermediate columns so the method stays reliable.
Students will:
- work outwards from the innermost operation in an expression
- use an intermediate column for each smaller part
- work out how many rows a truth table needs
- complete a truth table for a combined two-input expression
- complete a truth table for a three-input expression with brackets
Inside: 5 explanation cells, 2 multiple-choice questions, 2 fill-in-the-blanks cells and 2 written answers. 23 marks, about 45 minutes.
Series: J277 2.4 · Boolean logic, part 2 of 4.
Shared by Coding PathwayVerified teacher
- 11 cells
- About 45 minutes
- CC BY-SA 4.0
- Shared 17 Aug 2026
- Updated 9 Sept 2026
Preview
The whole resource, exactly as a class sees it. Answers and marking are held back.
Combined Boolean operators and truth tables
Logic systems often use more than one gate. The safest way to calculate a combined expression is to create an intermediate column for each smaller part, then work outward.
For the expression A AND NOT B:
- calculate
NOT B; - combine
Awith that result using AND.
Brackets make the intended grouping clear. Treat a bracketed section like a small circuit whose output becomes an input to the next gate.
Worked example: A AND NOT B
| A | B | NOT B | A AND NOT B |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 1 |
| 1 | 1 | 0 | 0 |
The only output of 1 occurs when A is 1 and B is 0. The NOT operation changes B before the AND gate uses it.
A OR NOT B.
| A | B | NOT B | A OR NOT B |
|---|---|---|---|
| 0 | 0 | label 1 | label 2 |
| 0 | 1 | label 3 | label 4 |
| 1 | 0 | label 5 | label 6 |
| 1 | 1 | label 7 | label 8 |
- 0
- 1
For A = 0 and B = 1, what is the output of NOT (A OR B)?
- A1
- B0
- CA
- DB
How many rows?
A truth table must include every possible input combination. Each extra Boolean input doubles the number of rows:
| Number of inputs | Required rows |
|---|---|
| 1 | 2 |
| 2 | 4 |
| 3 | 8 |
| 4 | 16 |
The rule is 2 raised to the number of inputs. Do not count intermediate columns as new inputs. For example, NOT B is calculated from B, so it does not double the number of rows.
A circuit has three independent inputs: A, B and C. How many rows must its complete truth table contain?
- A3
- B6
- C8
- D9
Three-input example: (A OR B) AND NOT C
Work from the two inner parts toward the final AND:
| A | B | C | A OR B | NOT C | Final output |
|---|---|---|---|---|---|
| 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 |
Writing intermediate columns reduces mental load and makes it easier to locate one incorrect step.
(A AND B) OR C. Work out A AND B before the final OR.
| A | B | C | Final output |
|---|---|---|---|
| 0 | 0 | 0 | label 1 |
| 0 | 0 | 1 | label 2 |
| 0 | 1 | 0 | label 3 |
| 0 | 1 | 1 | label 4 |
| 1 | 0 | 0 | label 5 |
| 1 | 0 | 1 | label 6 |
| 1 | 1 | 0 | label 7 |
| 1 | 1 | 1 | label 8 |
- 0
- 1
A greenhouse fan runs when the temperature is high AND the window is NOT open. Write a Boolean expression and explain when the fan output is 1.
Use clear input names or A and B, then translate each part of the sentence.
Students type their answer here.
Explain why intermediate columns are useful when completing a truth table with several operators.
Link each smaller operation to the final result and to checking work.
Students type their answer here.
Review
For every combined expression: list all input rows, calculate bracketed or NOT sections first, add an intermediate column for each gate, and calculate the final output last.