Boolean Logic

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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.

Simple logic circuit representation (series circuit analogy).

Truth table grid analogy (Punnett square style).

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?
Medium
- 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?
Medium
- 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?
Hard
- 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?
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 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?
Hard
- 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?
Medium
- 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?
Medium
- 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?
Medium
- 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?
Hard
- A0
- B1
- CCannot be determined
- DDepends on the gate technology
23.Which of the following truth tables represents a NOR gate?
Medium
- 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?
Medium
- 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?
Hard
- 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?
Hard
- 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?
Hard
- 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?
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 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?
Hard
- 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?
Hard
- 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?
Medium
- 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?
Medium
- 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?
Medium
- 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)

Slides

What is Boolean Logic?

Boolean logic is used in computer science and electronics to make logical decisions.
Boolean operators are either TRUE or FALSE, often represented as 1 or 0.
Inputs and outputs are given letters to represent them.
Special symbols are used to write expressions more easily.

Logic Gates Overview

Logic gates are a visual way of representing a Boolean expression.
The gates covered in this course are: AND, OR, NOT, XOR, NAND, NOR.
Each gate has a specific circuit symbol and truth table.

Which gate returns TRUE only if both inputs are TRUE?

AND Gate

Expression: (A AND B)
Returns TRUE only if both inputs are TRUE.
TRUE AND TRUE = TRUE; otherwise FALSE.

OR Gate

Expression: (A OR B)
Returns TRUE if either input is TRUE.
TRUE OR FALSE = TRUE; FALSE OR FALSE = FALSE.

What is the output of an OR gate when A=1 and B=0?

0
1
Cannot be determined
Depends on the gate

NOT Gate

Expression: (NOT A)
Reverses the input value.
NOT TRUE = FALSE; NOT FALSE = TRUE.

XOR Gate

Expression: (A XOR B)
Returns TRUE if either input is TRUE but NOT both.
TRUE XOR TRUE = FALSE; TRUE XOR FALSE = TRUE.

Which of the following is TRUE for XOR?

1 XOR 1 = 1
1 XOR 0 = 1
0 XOR 0 = 1
1 XOR 1 = 0

NAND and NOR Gates

NAND: (A NAND B) returns TRUE if either or both inputs are FALSE.
NOR: (A NOR B) returns TRUE only if both inputs are FALSE.
NAND is the opposite of AND; NOR is the opposite of OR.

A NAND gate is equivalent to a(n) ____ gate followed by a NOT gate.

Logic Circuits

A logic circuit performs logical operations on binary information.
Logic gates are the core components of logic circuits.
Brackets clarify the order of operations.
Circuits are limited to a maximum of three inputs and one output.

Example Logic Circuit

Expression: Q = NOT(A OR B)
Inputs A and B go into an OR gate.
The output of the OR gate goes into a NOT gate to produce Q.

In the circuit Q = NOT(A OR B), what is Q when A=1, B=0?

0
1
Cannot be determined
Depends on the gate

Truth Tables

A truth table shows all possible inputs and the corresponding outputs.
For n inputs, there are 2ⁿ rows.
Start with 000... and count up in binary to cover all combinations.

Truth Table for AND

Two inputs A and B, output A AND B.
Rows: 00→0, 01→0, 10→0, 11→1.

How many rows are in a truth table for three inputs?

Creating a Truth Table for a Circuit

For expression P = (A AND B) AND NOT C, there are 3 inputs (8 rows).
First compute intermediate columns like (A AND B) and NOT C.
Then combine them to get final output P.

For a truth table with 4 inputs, the number of rows is ____.

Logic Expressions

A logic expression is an equation representing a gate or circuit.
Output on the left, inputs and gates on the right.
Examples: Z = NOT A, Z = A AND B, Z = A OR B.

Which expression represents a NAND gate?

Z = A NAND B
Z = NOT (A AND B)
Z = A AND B
Both A and B

From Truth Table to Logic Circuit

Find rows where output = 1.
Build a logic branch for each such row.
Combine branches using OR gate.
Construct the logic expression and draw the circuit.

Which of these are true about Boolean logic?

Boolean values are TRUE or FALSE.
Inputs and outputs are represented by numbers.
Logic gates are a visual representation of Boolean expressions.
There are only three logic gates: AND, OR, NOT.

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