What is Boolean logic, really? Think of a light switch. It's either on or off — there's no "sort of on." Boolean logic is that same idea applied to reasoning: every statement is either completely TRUE or completely FALSE, nothing in between. Computers represent TRUE as 1 and FALSE as 0, and logic gates are the tiny electronic components that take these 1s and 0s as inputs and produce a 1 or 0 as output, based on a fixed rule.

Inputs and outputs are given single letters (A, B, C for inputs; often P, Q, X, Z for outputs) so we can write compact expressions instead of long sentences. A logic gate is just a visual symbol representing one rule — a picture standing in for a mini decision.

Imagine a car that will only start (output = 1) if both the key is turned and the seatbelt is on. Miss either one, and nothing happens. That's exactly how AND behaves — it's strict. It only outputs TRUE when every single input is TRUE.

Now imagine a doorbell that rings if you press the front button or the back button. You don't need both — just one is enough. OR is generous: it's TRUE as soon as at least one input is TRUE.

NOT is the simplest gate of all — it only has one input, and it just flips it. TRUE becomes FALSE, FALSE becomes TRUE. Think of it as a mirror.

This is the gate students mix up most often, so slow down here. XOR looks like OR but has one crucial difference: it is exclusive — meaning it wants exactly one input to be TRUE, not both. Picture two people trying to turn on a single hallway light using two separate switches at either end of the hallway — flipping just one switch turns the light on, but if both switches get flipped, the light effectively cancels out and turns off again.

NAND is simply AND with the output flipped — take whatever AND would output, then invert it. In the circuit symbol, this "invert" is drawn as a small bubble on the output of the AND shape.

Same idea again — NOR is OR with the result flipped. It's the "grumpiest" gate: the only way to get a TRUE output is if both inputs are FALSE.

A single gate is only useful for simple decisions. Real problems need multiple conditions chained together — that's a logic circuit: several gates wired up so the output of one gate becomes the input of the next. This is exactly like how in maths, you combine +, −, and × into one longer expression.

Brackets matter enormously. Just like (2 + 3) × 4 is different from 2 + (3 × 4), the expression (A AND B) OR C is a completely different circuit from A AND (B OR C). Always build the bracketed part first.

Reading a Circuit Diagram

Take the expression Q = NOT(A OR B). Here's how you'd read the circuit left to right:

Notice how P is just a label for the "in-between" wire — the output of the first gate and the input to the second. Only Q is the final output of the whole circuit. IGCSE circuits are capped at 3 inputs and 1 output, so you'll never have to track more wires than that.

Worked Example — Sprinkler System

"A sprinkler system switches on if it is not daytime (input A) and the temperature is greater than 40°C (input B)."

Step 1 — Translate the sentence into logic. "Not daytime" → NOT A. "and the temperature is greater than 40" → combine with AND. So: Q = (NOT A) AND B.

Step 2 — Build it gate by gate. A goes through a NOT gate first (because it needs inverting before use). That output, plus B untouched, both feed into an AND gate to produce Q.

This is a great example of the general method: read the sentence for connecting words ("and", "or", "not") — they tell you exactly which gates to use and in what order.

A truth table is the ultimate "show every possibility" tool. Instead of describing a rule in words, you just list every combination of inputs and write down what the output is for each one. It removes all ambiguity — nothing is left to interpretation.

Why 2ⁿ? Each input can independently be 0 or 1 — 2 choices. With 3 inputs, you're choosing a value for A (2 ways) AND B (2 ways) AND C (2 ways), so 2 × 2 × 2 = 8 total combinations. To make sure you don't accidentally skip a combination, count up in binary starting from all-zeros: 000, 001, 010, 011, 100, 101, 110, 111.

Worked Example — Building a Table for P = (A AND B) AND NOT C

The examiner-friendly method: build one column at a time, left to right, exactly following the brackets — never try to jump straight to the final answer.

Step 1: List all 8 rows for A, B, C.
Step 2: Add a column for the bracketed part first: A AND B.
Step 3: Add a column for NOT C.
Step 4: Combine those two new columns with AND to get the final answer, P.

Going Backwards — Circuit from a Truth Table

This is a favourite exam trick: they give you a completed truth table and ask you to build the circuit that produces it. Here's the reliable method, using this table for output X:

Step 1 — Find every row where the output is 1. Here, that's row (A=1, B=0, C=0) and row (A=1, B=1, C=1).

Step 2 — Build one "AND branch" per winning row. For each row, AND together the inputs exactly as they appear — using NOT on any input that's a 0 in that row.
• Row 1: A=1, B=0, C=0 → A AND (NOT B) AND (NOT C)
• Row 2: A=1, B=1, C=1 → A AND B AND C

Step 3 — OR the branches together. The final output is TRUE if either branch is true, so combine them with OR:

This method works every time, no matter how complicated the table looks — find the 1s, build an AND-branch for each, then OR them all together.

A logic expression is just the "equation" version of a circuit or truth table. The output goes on the left of the equals sign, and the inputs plus gates go on the right — exactly like how y = 2x + 3 describes a relationship using symbols instead of a graph.

For circuits with multiple gates, you build the expression by working through the diagram in the same order the signal flows — innermost brackets (the earliest gates) go inside the parentheses first.

Worked Example — Circuit to Expression

Consider a circuit where: A and B feed into a NAND gate, and separately B and C feed into a NOR gate; the NAND output goes through a NOT gate; then that inverted result and the NOR output both feed into a final OR gate producing X.

Build it piece by piece:

• NAND branch: A NAND B, then inverted → NOT(A NAND B)
• NOR branch: B NOR C
• Combine with the final OR gate:

The trick to these multi-gate questions is always the same: name each intermediate wire in your head (or on paper), work out its mini-expression, then plug that mini-expression into the next gate like a jigsaw piece.