Derive the truth table of a combinational logic circuit built from two or more gates and write its Boolean expression

What this dot point is asking

SEAB wants you to take a circuit made of two or more logic gates, work out its output for every possible input combination (its truth table), and write its Boolean expression. The central insight is that a combinational circuit has no memory: its output depends only on the inputs present right now, so you can work it out gate by gate using intermediate columns in a table.

The answer

What combinational logic is

A combinational logic circuit is built by connecting basic gates (AND, OR, NOT, NAND, NOR) so that the output depends only on the present combination of inputs. It has no memory of past inputs, unlike a latch or a counter. Because it has no memory, every behaviour is captured completely by a truth table listing the output for each input combination.

Listing the input combinations

For $n$ inputs there are $2^n$ combinations: two inputs give $4$ rows, three inputs give $8$ rows. Always list them in standard binary counting order ($00, 01, 10, 11$ for two inputs) so that no combination is forgotten and the table is easy to check.

Working through the gates with intermediate columns

The reliable method is to add a column for the output of each gate inside the circuit:

1. Write the input columns and list every combination in binary order.
2. Add a column for the first gate's output and fill it using that gate's rule.
3. Add columns for each later gate, working from the inputs towards the final output, filling each from the columns it depends on.
4. The last column is the circuit output $Q$.

These intermediate columns turn a confusing multi-gate circuit into a series of simple single-gate steps.

Writing the Boolean expression

The Boolean expression is built the same way, from the inputs outward. Replace each gate with its operator: AND becomes a dot, OR becomes a plus, NOT becomes a bar, and use brackets to show which signals feed which gate. For example, an AND fed by $\bar{A}$ and $B$ gives $Q = \bar{A} \cdot B$. The expression and the truth table describe the same circuit in two languages.

Examples in context

Example 1. A two-of-three voting circuit. A control system acts only when at least two of three sensors agree. Combining the sensors with AND and OR gates produces an output that is 1 whenever any two inputs are high. Deriving its eight-row truth table confirms the circuit behaves correctly before it is built, which is the everyday value of truth-table analysis.

Example 2. An exclusive-OR comparison. The circuit in the worked example outputs 1 only when its two inputs differ. This exclusive-OR behaviour is used to compare two bits and flag when they are not equal, a building block inside adders and error checkers. A few basic gates, analysed by truth table, deliver a genuinely useful function.

Try this

• Cue. State how many rows a truth table needs for a circuit with three inputs. Three inputs give $2^3 = 8$ combinations, so eight rows.

• Cue. Write the Boolean expression for a circuit where $A$ is inverted then ORed with $B$. The NOT gives $\bar{A}$, then OR with $B$: $Q = \bar{A} + B$.

• Cue. Explain why intermediate columns help when deriving a truth table. They break a multi-gate circuit into single-gate steps, so each column uses just one simple gate rule, reducing errors.