Boolean Logic

Overview

Boolean Logic is the foundation of digital circuits and computational logic. It involves expressions that evaluate to either True (1) or False (0), and it uses logical operators such as AND, OR, NOT, and XOR to form more complex conditions.

This note will guide you through defining problems using Boolean logic, constructing truth tables, and working with logic gates.

Boolean Operators and Truth Tables

AND (Conjunction, ∧):

Returns True if both inputs are True.

Truth Table:

A	B	A ∧ B
T	T	T
T	F	F
F	T	F
F	F	F

OR (Disjunction, ∨):

Returns True if at least one input is True.

Truth Table:

A	B	A ∨ B
T	T	T
T	F	T
F	T	T
F	F	F

NOT (Negation, ¬):

Returns the inverse of the input.

Truth Table:

A	¬A
T	F
F	T

XOR (Exclusive OR, ⊕):

Returns True if exactly one input is True.

Truth Table:

A	B	A ⊕ B
T	T	F
T	F	T
F	T	T
F	F	F

Constructing Logic Gate Diagrams from Boolean Expressions

Boolean expressions can be represented as logic gate diagrams. Here's a summary of the key gates:

AND Gate: Outputs True only if both inputs are True.
OR Gate: Outputs True if at least one input is True.
NOT Gate: Inverts the input.
XOR Gate: Outputs True if exactly one input is True.

lightbulbExample

Example Expression: $A \wedge (\neg B \vee C)$

Diagram:

First, use a NOT Gate for $B$ .
Then, use an OR Gate to combine $\neg B$ and $C$
Finally, use an AND Gate to combine $A$ with the result of ( $\neg B \vee C$ )

Constructing Boolean Expressions from Logic Gate Diagrams

Identify the gates and their connections.
Start from the inputs and move towards the output.
Combine sub-expressions for each gate.

lightbulbExample

Example Given a diagram with:

An AND Gate combining $A$ and $B$
The output of this AND Gate feeds into an OR Gate with input $C$ The corresponding Boolean expression is:

(A \wedge B) \vee C

Constructing Truth Tables

A truth table lists all possible combinations of input values and their corresponding output.

lightbulbExample

Example For the expression $A \wedge B$

A	B	$A \wedge B$
T	T	T
T	F	F
F	T	F
F	F	F

Examples of Problem Definition

Problem: Is the light on?

Logic: Light turns on when switch1 AND switch2 are ON.
Boolean Expression: $\text{switch1} \wedge \text{switch2}$

Problem: Car alarm activation.

Logic: Alarm activates if door open OR ignition ON while key not present.
Boolean Expression: $(\text{door} \vee \text{ignition}) \wedge \neg \text{key}$

Note Summary

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Common Mistakes

Incorrect Truth Table Construction:

Ensure all combinations of inputs are included.
Use the correct operator logic for each row.

Misinterpreting Logic Diagrams:

Follow the signal flow from the inputs to the output systematically.

Incorrect Use of XOR:

Confusing XOR with OR. XOR is true only when exactly one input is true.

Order of Operations:

Apply NOT before AND or OR in compound expressions.

infoNote

Key Takeaways

Boolean logic forms the basis of computational logic and circuit design.
Understand the behaviour of AND, OR, NOT, and XOR operators through truth tables.
Practice converting between Boolean expressions, truth tables, and logic gate diagrams.
Ensure accuracy by systematically constructing and validating truth tables and diagrams.

Boolean Logic Simplified Revision Notes for A-Level OCR Computer Science

Boolean Logic

Overview

Boolean Operators and Truth Tables

AND (Conjunction, ∧):

OR (Disjunction, ∨):

NOT (Negation, ¬):

XOR (Exclusive OR, ⊕):

Constructing Logic Gate Diagrams from Boolean Expressions

Constructing Boolean Expressions from Logic Gate Diagrams

Constructing Truth Tables

Examples of Problem Definition

Note Summary

Common Mistakes

Key Takeaways

500K+ Students Use These Powerful Tools to Master Boolean Logic For their A-Level Exams.

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