Use Boolean algebra, logic gates and truth tables to represent and simplify logical operations.

What this dot point is asking

WJEC wants you to know the logic gates and their truth tables, to write and evaluate Boolean expressions, and to simplify them using Boolean identities. Logic is the bridge between binary data and the physical circuits of the processor, so this topic links data representation to hardware. Expect a truth-table question and a simplification question on most AS papers, both of which reward careful, systematic working.

The answer

The logic gates

Each gate has a standard symbol and a truth table. NAND and NOR are called universal gates because any logic function can be built from NAND gates alone, or from NOR gates alone, which matters when manufacturing a chip from a single gate type.

Truth tables

A truth table lists every possible combination of the inputs and the resulting output. For n inputs there are 2 to the power n rows: two inputs give four rows, three inputs give eight. Building a truth table for a compound expression means adding a column for each intermediate part and working left to right, evaluating the innermost brackets first.

Boolean algebra and identities

De Morgan's laws are the workhorse of simplification because they let you push a NOT through a bracket, turning an AND into an OR and vice versa. They are also how a circuit built only from NAND or NOR gates is derived.

Simplifying a circuit

Fewer gates means lower cost, lower power and faster operation, so simplification is not just algebra for its own sake: it is how real circuits are made efficient.

Examples in context

Example 1. A half adder built from gates

Adding two single bits needs a sum and a carry. The sum is A XOR B (1 when the bits differ) and the carry is A AND B (1 only when both are 1). This half adder is the smallest piece of the arithmetic unit, and it shows how the XOR and AND gates combine to do real binary arithmetic inside the processor.

Example 2. A safety interlock on a machine

A press should only operate when a guard is closed (G) AND a start button is pressed (S) AND an emergency stop is not active (NOT E). The control logic is G AND S AND (NOT E), a direct Boolean expression that a small circuit implements, illustrating how logic gates enforce real-world safety rules.

Example 3. Cutting gate count to save cost

A design originally uses the expression (A AND B) OR (A AND C), which needs two AND gates and an OR gate. The distributive law rewrites it as A AND (B OR C), needing one AND gate and one OR gate. On a chip with millions of such blocks, this kind of simplification reduces silicon area, power draw and propagation delay, which is why examiners stress it.

Try this

Q1. Write down the truth table output column for A NAND B for the input rows 00, 01, 10, 11. [2 marks]

• Cue. NAND is NOT AND, so the outputs are 1, 1, 1, 0.

Q2. Use De Morgan's law to rewrite NOT (A OR B). [1 mark]

• Cue. (NOT A) AND (NOT B).