Boolean Logic

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Lesson notes

Boolean Logic Basics

Boolean logic is used in computer science and electronics to make logical decisions.
Boolean values are either TRUE or FALSE, often represented as 1 or 0.
Inputs and outputs are given letters (e.g., A, B, Q).
Special symbols are used to write Boolean expressions concisely.

Logic Gates Overview

Logic gates are visual representations of Boolean expressions.
The six gates covered are: AND, OR, NOT, XOR, NAND, NOR.
Each gate has a unique circuit symbol and truth table.
Gates can be combined to form logic circuits.

AND, OR, NOT Gates

AND: Output TRUE only if both inputs are TRUE. Expression: A AND B.
OR: Output TRUE if at least one input is TRUE. Expression: A OR B.
NOT: Output is the opposite of the input. Expression: NOT A.
Truth tables list all input combinations and the corresponding output.

XOR, NAND, NOR Gates

XOR (exclusive OR): Output TRUE if exactly one input is TRUE. Expression: A XOR B.
NAND (NOT AND): Output FALSE only if both inputs are TRUE. Expression: A NAND B.
NOR (NOT OR): Output TRUE only if both inputs are FALSE. Expression: A NOR B.
Common exam mistake: confusing OR and XOR. XOR is false when both inputs are TRUE.

Logic Circuits

A logic circuit performs logical operations using multiple gates.
Logic diagrams show the arrangement of gates and connections.
Brackets clarify the order of operations in expressions.
Circuits are limited to a maximum of three inputs and one output.
Draw circuits from left to right, labelling all inputs clearly.

Truth Tables for Circuits

A truth table lists all possible inputs and the resulting output.
Number of rows = 2ⁿ, where n is the number of inputs.
List inputs in binary order starting from 000.
Add intermediate columns for sub-expressions if needed.
The final column shows the output of the whole expression.

Creating Circuits from Truth Tables

Identify rows where output = 1.
For each such row, create a logic branch using AND and NOT gates.
Combine all branches with an OR gate.
Write the Boolean expression and draw the circuit.
Example: (A AND NOT B AND NOT C) OR (A AND B AND C).

Logic Expressions

A logic expression is an equation: output = function of inputs.
Each gate has a standard expression (e.g., Z = A AND B).
Complex circuits yield expressions like Q = NOT(A OR B).
You may be asked to write an expression from a circuit or truth table, or vice versa.

Slides

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Practice questions

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1.Which logic gate returns TRUE only when both inputs are TRUE?
Easy
- AAND
- BOR
- CXOR
- DNAND
2.What is the output of a NOT gate when the input is 1?
Easy
- A0
- B1
- CBoth 0 and 1
- DUndefined
3.Which of the following truth tables represents an XOR gate?
Easy
- AA B Q\n0 0 0\n0 1 1\n1 0 1\n1 1 0
- BA B Q\n0 0 0\n0 1 1\n1 0 1\n1 1 1
- CA B Q\n0 0 1\n0 1 1\n1 0 1\n1 1 0
- DA B Q\n0 0 1\n0 1 0\n1 0 0\n1 1 0
4.How many rows are needed in a truth table for a logic circuit with three inputs?
Easy
- A3
- B6
- C8
- D9
5.What is the Boolean expression for the logic circuit that has an OR gate with inputs A and B, followed by a NOT gate?
Easy
- ANOT(A OR B)
- BA OR NOT B
- CNOT A OR B
- DA NOR B
6.Which gate is represented by the symbol that looks like an OR gate with a small circle at the output?
Easy
- ANOR
- BNAND
- CXOR
- DNOT
7.For the Boolean expression P = (A AND B) AND NOT C, what is the value of P when A=1, B=1, C=1?
Medium
- A0
- B1
- CCannot be determined
- DDepends on the gate
8.Which of the following Boolean expressions corresponds to the truth table where X=1 only when A=1, B=0, C=0 OR when A=1, B=1, C=1?
Medium
- A(A AND NOT B AND NOT C) OR (A AND B AND C)
- B(A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- C(A OR B OR C) AND (NOT A OR NOT B OR NOT C)
- D(A AND B) OR (NOT B AND NOT C)

More questions

9.In Boolean logic, what do the values 1 and 0 typically represent?
Easy
- ATRUE and FALSE
- BHIGH and LOW voltage
- CON and OFF
- DAll of the above
10.What is the output of a NAND gate when both inputs are 0?
Easy
- A1
- B0
- CUndefined
- DDepends on the order
11.A logic circuit has two inputs A and B. The output Q = (NOT A) AND B. Which combination of inputs gives Q=1?
Easy
- AA=0, B=1
- BA=1, B=0
- CA=0, B=0
- DA=1, B=1
12.Which of the following is NOT a logic gate covered in the IGCSE Computer Science syllabus?
Easy
- AAND
- BOR
- CXOR
- DXNOR
13.How many rows are needed in a truth table for a logic circuit with 3 inputs?
Easy
- A3
- B6
- C8
- D9
14.A logic circuit has three inputs A, B, C. The output X is 1 only when A=1, B=0, C=0 OR when A=1, B=1, C=1. Which Boolean expression represents X?
Medium
- AX = (A AND NOT B AND NOT C) OR (A AND B AND C)
- BX = (A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- CX = (A OR B OR C) AND (NOT A OR NOT B OR NOT C)
- DX = (A AND B AND C) OR (A AND NOT B AND NOT C)
15.What is the output of a NAND gate when both inputs are 1?
Easy
- A0
- B1
- CUndefined
- DDepends on the gate
16.A logic circuit has three inputs A, B, C. The output X is 1 when A=1, B=0, C=0 OR when A=1, B=1, C=1. Which Boolean expression represents X?
Medium
- AX = (A AND NOT B AND NOT C) OR (A AND B AND C)
- BX = (A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- CX = (A OR NOT B OR NOT C) AND (A OR B OR C)
- DX = (A AND B AND C) OR (A AND NOT B AND C)
17.What is the output of a NOR gate when both inputs are 1?
Easy
- A0
- B1
- CCannot be determined
- DIt depends on the type of NOR gate
18.A student builds a logic circuit with three inputs. They want the output to be 1 only when exactly two of the inputs are 1. Which combination of gates would produce this?
Hard
- AUse an XOR gate for two inputs, then AND with the third input
- BUse an AND gate for two inputs, then OR with the third input
- CUse a NAND gate for all three inputs
- DUse a three-input XOR gate
19.Given the truth table below, what is the simplified Boolean expression for X? A B C | X 0 0 0 | 0 0 0 1 | 0 0 1 0 | 0 0 1 1 | 1 1 0 0 | 0 1 0 1 | 1 1 1 0 | 0 1 1 1 | 1
Hard
- AX = (A AND C) OR (B AND C)
- BX = (A AND B) OR (B AND C)
- CX = (A AND C) OR (A AND B)
- DX = (A OR B) AND C
20.Which logic gate is represented by the symbol that looks like an AND gate with a small circle at its output?
Easy
- ANAND
- BNOR
- CXOR
- DNOT
21.A security system has two sensors: A (door) and B (window). The alarm (X) should sound if the door is opened (A=1) OR the window is opened (B=1). However, there is a master switch C that disables the alarm when C=1 (disabled). Which Boolean expression represents the alarm output?
Medium
- AX = (A OR B) AND NOT C
- BX = (A AND B) OR NOT C
- CX = (A OR B) OR C
- DX = (A AND B) AND NOT C
22.A logic circuit has inputs A, B, C and output Q. The Boolean expression is Q = (A NAND B) NOR C. What is the value of Q when A=1, B=1, C=1?
Medium
- A0
- B1
- CCannot be determined
- DDepends on the gate technology
23.Which of the following truth tables represents a NOR gate?
Easy
- AA B | Q 0 0 | 1 0 1 | 0 1 0 | 0 1 1 | 0
- BA B | Q 0 0 | 1 0 1 | 1 1 0 | 1 1 1 | 0
- CA B | Q 0 0 | 0 0 1 | 1 1 0 | 1 1 1 | 0
- DA B | Q 0 0 | 0 0 1 | 0 1 0 | 0 1 1 | 1
24.What is the output of a XOR gate when both inputs are 1?
Easy
- A0
- B1
- CCannot be determined
- DIt depends on the voltage levels
25.A logic circuit is built from three gates: an OR gate with inputs A and B, its output goes into a NAND gate with the other input being C. The output of the NAND gate is the final output Q. Which Boolean expression represents Q?
Medium
- AQ = NOT ((A OR B) AND C)
- BQ = (A OR B) NAND C
- CQ = (A OR B) AND NOT C
- DQ = NOT (A OR B) NAND C
26.A security system has two sensors: A (door) and B (window). The alarm (X) should sound if the door is opened (A=1) OR the window is opened (B=1). However, there is a master switch C that disables the alarm when C=1. Which Boolean expression represents the alarm output?
Medium
- AX = (A OR B) AND NOT C
- BX = (A OR B) OR NOT C
- CX = (A AND B) AND NOT C
- DX = (A OR B) AND C
27.A logic circuit is built from three gates: an OR gate with inputs A and B, its output goes into a NAND gate with the other input being C. The output of the NAND gate is Q. Which Boolean expression represents Q?
Medium
- AQ = NOT((A OR B) AND C)
- BQ = (A OR B) NAND C
- CQ = (A OR B) AND NOT C
- DQ = NOT(A OR B) AND C
28.A student builds a logic circuit with three inputs (A, B, C) and one output (X). The truth table for X shows that X = 1 only when A = 1, B = 0, C = 0 OR when A = 1, B = 1, C = 1. Which Boolean expression represents X?
Hard
- A(A AND NOT B AND NOT C) OR (A AND B AND C)
- B(A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- C(A OR NOT B OR NOT C) AND (A OR B OR C)
- D(A AND B AND C) OR (A AND NOT B AND C)
29.A logic circuit has three inputs A, B, C and output Q. The Boolean expression is Q = (A NAND B) NOR C. What is the value of Q when A=1, B=1, C=1?
Medium
- A0
- B1
- CCannot be determined
- DDepends on gate delays
30.Which logic gate gives an output of 1 only when both inputs are 0?
Easy
- ANOR
- BNAND
- CAND
- DOR
31.A security system has two sensors: A (door) and B (window). The alarm (X) should sound if the door is opened (A=1) OR the window is opened (B=1). However, there is a master switch C that disables the alarm when C=0. Which Boolean expression represents X?
Medium
- A(A OR B) AND C
- B(A OR B) OR C
- C(A AND B) AND C
- D(A OR B) AND NOT C
32.A logic circuit is built from three gates: an OR gate with inputs A and B, its output goes into a NAND gate with the other input being C. The output of the NAND gate is the final output Q. Which truth table correctly represents Q?
Medium
- AA B C Q: 000=1, 001=1, 010=1, 011=0, 100=1, 101=1, 110=1, 111=0
- BA B C Q: 000=0, 001=0, 010=0, 011=1, 100=0, 101=0, 110=0, 111=1
- CA B C Q: 000=1, 001=1, 010=1, 011=1, 100=1, 101=1, 110=1, 111=0
- DA B C Q: 000=1, 001=0, 010=1, 011=0, 100=1, 101=0, 110=1, 111=0
33.How many rows are needed in a truth table for a logic circuit with 4 inputs?
Easy
- A16
- B8
- C4
- D12
34.Which of the following truth tables represents a NAND gate?
Easy
- AA B Q: 00=1, 01=1, 10=1, 11=0
- BA B Q: 00=0, 01=0, 10=0, 11=1
- CA B Q: 00=1, 01=0, 10=0, 11=0
- DA B Q: 00=0, 01=1, 10=1, 11=1
35.A logic circuit has inputs A and B. The output Q = (NOT A) AND B. Which combination of inputs gives Q=1?
Easy
- AA=0, B=1
- BA=1, B=0
- CA=0, B=0
- DA=1, B=1
36.Consider the logic statement: X = (((A AND B) OR (C AND NOT B)) XOR NOT C). What is the value of X when A=1, B=1, C=0?
Hard
- A0
- B1
- CCannot be determined
- DDepends on gate delays
37.A student is asked to draw a logic circuit for the expression (A AND B) OR (NOT C). Which of the following describes the correct arrangement of gates?
Medium
- AAn AND gate with inputs A and B, a NOT gate on input C, then an OR gate combining the outputs of the AND and NOT gates.
- BAn OR gate with inputs A and B, a NOT gate on input C, then an AND gate combining the outputs of the OR and NOT gates.
- CA NOT gate on input C, then an AND gate with inputs A and B, then an OR gate combining the outputs of the AND and NOT gates.
- DAn AND gate with inputs A and B, then a NOT gate on the output of the AND, then an OR gate with input C.
38.A logic circuit has three inputs A, B, C and output X. The truth table is given below. Which Boolean expression correctly represents X? A B C | X 0 0 0 | 0 0 0 1 | 0 0 1 0 | 0 0 1 1 | 1 1 0 0 | 0 1 0 1 | 1 1 1 0 | 0 1 1 1 | 0
Hard
- A(NOT A AND B AND C) OR (A AND NOT B AND C)
- B(A AND B AND C) OR (A AND NOT B AND C)
- C(NOT A AND B AND C) OR (A AND B AND NOT C)
- D(A AND B AND C) OR (NOT A AND NOT B AND C)
39.What is the output of a NOR gate when both inputs are 0?
Easy
- A0
- B1
- CCannot be determined
- DDepends on the gate
40.A logic circuit has three inputs A, B, and C. The output X is 1 only when A=1, B=0, C=0 OR when A=1, B=1, C=1. Which Boolean expression represents X?
Hard
- A(A AND NOT B AND NOT C) OR (A AND B AND C)
- B(A OR NOT B OR NOT C) AND (A OR B OR C)
- C(A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- DA AND (B XOR C)
41.Which logic gate returns TRUE only when both inputs are FALSE?
Easy
- ANOR
- BNAND
- CXOR
- DOR
42.Boolean logic uses the values TRUE and FALSE, often represented as 1 and 0 respectively.
Easy
True or false?
43.Match each logic gate to its correct truth table output when both inputs are 1.
Easy
- AND
- NAND
- XOR
- OR
- 0
- 1
- 0
- 1
44.Arrange the following steps in the correct order to create a truth table for a logic circuit with three inputs.
Medium
- List all possible input combinations
- Calculate the number of rows needed (2ⁿ)
- Add columns for intermediate outputs
- Add final output column
45.A truth table for a logic circuit with 3 inputs requires 6 rows.
Easy
True or false?
46.Which Boolean expression represents a logic circuit that outputs 1 only when input A is 1 and input B is 0?
Easy
- AA AND NOT B
- BNOT A AND B
- CA OR NOT B
- DNOT A OR B
47.Which of the following are valid logic gates covered in the IGCSE Computer Science syllabus? (Select all that apply)
Easy
- AAND
- BOR
- CXOR
- DXNOR
- ENAND
48.A logic circuit has inputs A, B, C and output X. The truth table shows X = 1 only when (A=1, B=0, C=0) or (A=1, B=1, C=1). What is the Boolean expression for X?
Hard
- AX = (A AND NOT B AND NOT C) OR (A AND B AND C)
- BX = (A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- CX = (A OR B OR C) AND (NOT A OR NOT B OR NOT C)
- DX = (A AND B) OR (NOT B AND C)
49.A student builds a logic circuit with inputs A, B, C and output X. The truth table shows X = 1 only when A=1, B=0, C=0 or when A=1, B=1, C=1. Which Boolean expression represents X?
Hard
- AX = (A AND NOT B AND NOT C) OR (A AND B AND C)
- BX = (A AND B AND C) OR (NOT A AND NOT B AND NOT C)
- CX = (A OR B OR C) AND (NOT A OR NOT B OR NOT C)
- DX = (A AND B) OR (NOT B AND C)
50.Which of the following are logic gates covered in the IGCSE Computer Science syllabus? (Select all that apply)
Easy
- AAND
- BXOR
- CNAND
- DXNOR
51.Complete the sentence.
Easy
A logic gate that outputs 1 only when both inputs are 0 is called a ____ gate.
52.Write the Boolean expression for the logic circuit shown: an OR gate with inputs A and B, followed by a NOT gate.
Medium
53.How many rows are needed in a truth table for a logic circuit with 2 inputs?
Easy
54.Complete the truth table for a NAND gate with inputs A and B. Write the output for each row.
Medium
A=0, B=0 → output = ____ A=0, B=1 → output = ____ A=1, B=0 → output = ____ A=1, B=1 → output = ____
55.Write the Boolean expression for the logic circuit shown: an AND gate with inputs A and B, followed by a NOT gate.
Medium
56.What is the output of a XOR gate when both inputs are 0?
Easy
57.Explain the difference between an OR gate and an XOR gate.
Medium
58.How many rows are needed in a truth table for a logic circuit with 2 inputs?
Easy
- A4
- B5
- C6
- D3

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