How Does a Truth Table Relate to a Boolean Expression?

A truth table lists the output of a Boolean expression for every possible combination of input values. Therefore, the expression states the logic symbolically, while the truth table demonstrates its complete input-output behaviour.

How the Relationship Works

A Boolean expression combines binary variables using logical operators such as NOT, AND, and OR. For example, consider:

$$F = A \land \lnot B$$

Here, $F$ is true only when $A$ is true and $B$ is false. Its truth table is:

With $n$ input variables, a complete truth table contains $2^n$ rows. Since this example has two inputs, it requires $2^2 = 4$ rows.

The reverse process is often expressed in sum-of-products form. For each row where $F = 1$, include every input variable in an AND term: use the variable itself when its value is $1$ and its negation when its value is $0$. Then OR the row terms together. If several rows produce $1$, this first expression may be simplified using Boolean laws without changing its truth table. This provides a systematic expression directly from the output column.

For the table above, the only output of $1$ occurs when $A = 1$ and $B = 0$, producing $A \land \lnot B$.

A common misconception is that each truth table has only one possible expression. In fact, different but logically equivalent expressions can produce the same output column. For example, $A \land \lnot B$ and $\lnot(B \lor \lnot A)$ have identical truth tables.

Exam Technique

For A1.2 Data representation and computer logic, an examiner may ask you to construct a truth table or determine an expression from one. Include every input combination, show useful intermediate columns, and apply NOT before AND, then OR, unless brackets change the order.