Number Of Rows

A truth table must include every possible combination of input values. With one input there are 2 rows. With two inputs there are 4 rows. With three inputs there are 8 rows. This follows 2 raised to the number of inputs.

The syllabus limits logic circuits to a maximum of three inputs, so candidates need truth tables of at most eight input rows. The heading row is not counted as an input combination.

If an expression uses the same variable more than once, it is still one input variable. For example, (A AND B) OR (NOT A) has two inputs, A and B, and therefore four rows.

Standard Input Order For Two Inputs

Standard Input Order For Three Inputs

A Reliable Pattern Method

For three inputs, A changes least often: four 0s followed by four 1s. B changes every two rows: 0,0,1,1 repeated. C changes every row: 0,1 repeated.

This pattern ensures that no combination is missed or duplicated. It also matches the example ordering given in the Cambridge syllabus.

If a question already provides input rows in a different order, do not rearrange them; calculate the output for the rows as printed.

Using Intermediate Columns

For X = (A AND B) OR (NOT C), use one column P = A AND B, another column Q = NOT C, and then the final column X = P OR Q.

Each row should be processed gate by gate. Intermediate columns reduce mental load and reveal where an error occurred.

The intermediate column headings should state the exact operation so that the marker can follow the method.

Worked Table Structure

Truth Table From A Circuit

Follow the circuit from left to right. Give each gate output a temporary label. Add a table column for each labelled result. The final column is the circuit output.

If two gates are independent and both feed a final gate, calculate both intermediate outputs before the final output. Do not calculate the final gate using original inputs unless the wires connect directly.

A branch copies a signal and does not require a new independent input column.

Truth Table From A Problem Statement

First define each input variable and when it equals 1. Translate the statement into an expression or circuit. Then generate all input rows and evaluate the rule.

For example, a warning sounds when both A and B are true or when C is not true. This gives X = (A AND B) OR (NOT C), which can be evaluated using the table shown.

Checking the statement against selected rows is a useful verification step. For A = 0, B = 0, C = 0, the NOT C condition is true, so the warning should be 1.

Checking A Completed Table

Confirm the correct row count, confirm no repeated input combinations, check each intermediate column against its gate truth table, and finally verify a few output rows directly from the original statement.

For a two-input gate, the output pattern can also be compared with known patterns: AND 0001, OR 0111, NAND 1110, NOR 1000, XOR 0110 in standard 00,01,10,11 order.

Worked Examples

Examination Guidance

• Use 2^n to check the number of input rows.
• Write three-input rows in the 000 to 111 order unless another order is supplied.
• Add one intermediate column for each gate stage.
• Calculate from the innermost operation or earliest circuit gate outward.
• Check the final table against the original statement, not only the intermediate arithmetic.

Knowledge Check

1. How many rows are needed for two inputs?

2. How many rows for three inputs?

3. What is the standard three-input order?

4. Why use intermediate columns?

5. Does a branched wire create a new input?