AS · Practice questions · Logic circuits, truth tables and expressions

Every way round.

Six original Cambridge-style questions on constructing logic circuits, truth tables and expressions from each other and from problem statements, with mark-scheme style solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

How to earn the marks

Bracket by bracket.

Expression to circuit: one gate per bracket, two inputs per gate (except NOT), label inputs and output.
Truth table: rows 000 to 111 in order, a working column for each intermediate gate.
Statement to expression: one letter per condition, brackets around conditions that belong together.
Table to expression: one AND term per row with X = 1, joined with OR; check one row of each output.

Apply

[4 marks]

A logic circuit has a NAND gate with inputs A and B, and an OR gate with inputs B and C. The outputs of these two gates are the inputs of an XOR gate, whose output is X.

(a) Write the logic expression for the circuit.
(b) Complete the output column X for the inputs A B C = 000, 001, 010, 011, 100, 101, 110, 111.

(a) X = (A NAND B) XOR (B OR C). ✓
(b) Working: A NAND B = 1, 1, 1, 1, 1, 1, 0, 0 and B OR C = 0, 1, 1, 1, 0, 1, 1, 1. ✓
Rows 000 to 011: X = 1, 0, 0, 0. ✓
Rows 100 to 111: X = 1, 0, 1, 1. ✓

Write

[3 marks]

A car's warning buzzer W sounds if the engine is running (E = 1) and the seat belt is not fastened (B = 0), or if the engine is running and a door is open (D = 1). Write a logic expression for W.

(E AND NOT B) for the seat-belt condition. ✓
(E AND D) for the door condition. ✓
W = (E AND NOT B) OR (E AND D). (Equivalent: W = E AND (NOT B OR D).) ✓

Apply

[4 marks]

Complete the truth table for the car buzzer in the previous question, for the inputs E B D = 000 to 111.

Rows 000, 001, 010, 011 (engine off): W = 0, 0, 0, 0. ✓
Row 100: W = 1; row 101: W = 1. ✓
Row 110: W = 0 (belt fastened, door closed). ✓
Row 111: W = 1 (door open). ✓

Write

[3 marks]

A truth table for inputs A, B and C has the output X = 1 in two rows only: A B C = 001 and A B C = 110. Write a logic expression for X.

Term for 001: (NOT A AND NOT B AND C). ✓
Term for 110: (A AND B AND NOT C). ✓
Terms joined by OR: X = (NOT A AND NOT B AND C) OR (A AND B AND NOT C). ✓

Trace

[3 marks]

X = (A XOR B) NAND (NOT C). Find X for each set of inputs, showing your working: (a) A = 1, B = 1, C = 0; (b) A = 1, B = 0, C = 0; (c) A = 0, B = 1, C = 1.

(a) A XOR B = 0, NOT C = 1, 0 NAND 1 = 1. ✓
(b) A XOR B = 1, NOT C = 1, 1 NAND 1 = 0. ✓
(c) A XOR B = 1, NOT C = 0, 1 NAND 0 = 1. ✓

Analysis

[3 marks]

A student draws the circuit for X = NOT (A AND B) OR (B XOR C) using a single three-input gate and no NOT gate. Describe a correct circuit and give the only input row for which X = 0.

An AND gate with inputs A and B, whose output goes into a NOT gate. ✓
An XOR gate with inputs B and C; the outputs of the NOT gate and the XOR gate are the two inputs of an OR gate, giving X. ✓
X = 0 only for A = 1, B = 1, C = 1 (NOT (1 AND 1) = 0 and 1 XOR 1 = 0). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.