Simplifying Boolean Expressions and Logic Tables

Simplify Boolean expressions by applying one rule at a time. Complete logic tables, also called truth tables, by listing every possible input combination and calculating intermediate results before the final output. This makes errors easier to spot.

Boolean logic rules

A Boolean value has only two states: $1$ means true and $0$ means false. A logic operator combines or changes these values according to a fixed rule.

• $NOTA$ reverses the value of $A$.
• $AANDB$ is true only when both inputs are true.
• $AORB$ is true when either input, or both inputs, are true.

Useful simplification rules include:

$$AAND1=A$$ $$AOR0=A$$ $$AAND0=0$$ $$NOT(NOTA)=A$$

These rules show that an input remains unchanged when it is combined with the neutral value. A value that forces a fixed result, such as false with AND, is sometimes called a dominating value.

Worked example

For example, simplify:

$$(AAND1)OR(AAND0)$$

First, the first bracket becomes $A$, while the second bracket becomes $0$. The expression is therefore:

$$AOR0=A$$

The final answer is $A$. This simplification works for both possible values of $A$, so a truth table would give the same outputs as the original expression.

Completing a logic table

Consider:

$$Q=(AANDB)OR\; NOTA$$

With two inputs, there are four possible input combinations. List them systematically and use one intermediate column for each operation. This prevents a mistake in an early stage from being hidden inside the final answer.

Read each row from left to right. For example, when $A=0$ and $B=1$, the AND result is false, but NOT $A$ is true, so the final OR result is true.

Order and exam tip

Work inside brackets first, then apply $NOT$, followed by $AND$ and finally $OR$, unless brackets change the order. Keep intermediate columns clearly labelled and check that every input combination appears exactly once.

Common exam mistake: treating $OR$ as “one but not both”. In Boolean logic, $1OR1$ gives $1$, not $0$. OR is inclusive, meaning both true inputs still produce a true output. Do not confuse it with exclusive OR, which follows a different rule.