Logic Expressions and Circuits

Syllabus & Goals 3 min

Cambridge 10.1 · Logic expressions ↔ logic circuits Paper 2 · Algorithms, Programming and Logic

By the end of this lesson you can:

1. Write a logic expression from a logic circuit.
2. Draw a logic circuit from a logic expression, using only the six gates.
3. Complete a truth table directly from a logic expression.

Textbook: Chapter 10, §10.3 Examples 2, 3 and 8 (pp. 363–364, 366–367, 375–376)

Recap / Warm-Up 5 min

Last lesson you filled truth tables from circuits, one working column per gate. Each of those columns was really a small expression.

Quick starter

A circuit has an AND gate fed by A and B. Its output goes into a NOT gate. Write the output as one expression.

Key Concept 14 min

1 · A logic expression is a circuit written in words

A logic expression uses the gate names and brackets, for exampleX = (A OR NOT B) NAND (B XOR C). Every pair of brackets is one gate. The operator that sits outside every bracket is the final gate, nearest the output.

2 · Circuit → expression

1. Find the gates connected directly to the inputs. Write each as a bracket, e.g. (A NOR B). These are the innermost brackets.
2. Move right. For each gate, replace its inputs with the brackets that feed it. This nests the brackets in the right order.
3. The last gate joins everything: X = (…) OR (…). Its operator is the one outside all brackets.

3 · Expression → circuit

1. Find the operator outside every bracket. Draw that gate on the right, with output X. It is the last thing the circuit does.
2. Each bracket becomes the gate feeding one of its inputs. Work backwards from the output towards the inputs.
3. NOT applies only to the next input or bracket. Draw a NOT gate on that wire. NOT B is a NOT gate on B alone.
4. If an input is used twice, split its wire with a junction dot. Examiners expect one labelled input line per letter.

4 · Brackets change the meaning

Worked Example 12 min

(a) Write the logic expression for this circuit

1. Gate 1 takes A and B: (A NOR B). It is connected straight to the inputs.
2. Gate 2 takes B and C: (B AND C). The dot on B shows it feeds gates 1 and 2.
3. Gate 3 takes gate 2 and input C: ((B AND C) XOR C). C reaches gate 3 directly, below gate 2.
4. Gate 4 joins gates 1 and 3 with OR. This is the final gate.

X = (A NOR B) OR ((B AND C) XOR C)

Marks are usually given for each correct part: gate 1, gate 3 (which contains gate 2) and the final OR.

(b) Draw the circuit and truth table for X = (A OR NOT B) NAND (B XOR C)

1. The operator outside the brackets is NAND. Draw a NAND gate with output X. It joins the two brackets.
2. Left bracket: an OR gate with inputs A and NOT B. NOT B needs its own NOT gate on the B wire.
3. Right bracket: an XOR gate with inputs B and C. B is used twice, so its wire splits.
4. Fill the truth table with columns NOT B, P, Q and X. One column per gate, as in Lesson 3.

Try It Yourself 12 min

📝 Exam Practice 10 min

Recap & Key Terms 3 min

Write an expression by working from the inputs to the output, one bracket per gate. Draw a circuit by working from the output back: the operator outside all brackets is the last gate. NOT applies only to what follows it.

Logic expression

A statement of a circuit's function using gate names and brackets, e.g. X = (A AND B) OR NOT C.

Final (output) gate

The gate whose operator lies outside all brackets; it produces X.

Brackets

Group the inputs of one gate; each bracketed part is one gate's output.

Boolean algebra

A symbolic form of logic (. for AND, + for OR, a bar for NOT); beyond the IGCSE syllabus.

Homework 1 min

Task (≤ 15 min): For X = (NOT A AND B) NOR C: (a) draw the logic circuit; (b) complete the truth table with working columns.