Simplifying Boolean Algebra

Overview

Boolean Algebra is a mathematical framework used to simplify and manipulate logical expressions. In Computer Science, it's essential for designing efficient logic circuits and optimising algorithms. Simplifying Boolean expressions reduces complexity, saves resources, and ensures faster processing. This note focuses on key rules: De Morgan's Laws, Distribution, Association, Commutation, and Double Negation, to help you simplify Boolean statements.

De Morgan's Laws

These laws allow the transformation of expressions involving negations and logical operators. They state:

\neg(A \land B) = \neg A \lor \neg B

\neg(A \lor B) = \neg A \land \neg B

Explanation: These rules flip the logical operator and negate each operand.

Distribution

The distributive property enables restructuring expressions involving $\land$ (AND) and $\lor$ (OR):

Distribute OR over AND:

A \lor (B \land C) = (A \lor B) \land (A \lor C)

Distribute AND over OR:

A \land (B \lor C) = (A \land B) \lor (A \land C)

Explanation: This rule allows factoring expressions, much like distributing multiplication over addition in algebra.

Association

This rule states that the grouping of operands doesn't affect the result:

(A \lor B) \lor C = A \lor (B \lor C)

(A \land B) \land C = A \land (B \land C)

Explanation: You can rearrange the parentheses without changing the outcome.

Commutation

Commutation allows operands to be swapped:

A \lor B = B \lor A

A \land B = B \land A

Explanation: Order doesn't matter in logical OR and AND operations.

Double Negation

This rule states that negating a negation returns the original value:

\neg(\neg A) = A

Explanation: Cancelling out two negations results in the initial value.

Examples

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Example 1: Simplify $\neg(A \land B) \lor A$

Step 1: Apply De Morgan's Law:

\neg A \lor \neg B \lor A

Step 2: Apply Commutation:

(\neg A \lor A) \lor \neg B

Step 3: Simplify using $A \lor \neg A = 1$

1 \lor \neg B

Step 4: Simplify further:

Result = 1

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Example 2: Simplify $(A \lor B) \land (A \lor \neg B)$

Step 1: Apply Distribution:

(A \land (A \lor \neg B)) \lor (B \land (A \lor \neg B))

Step 2: Simplify terms using Absorption

(e.g., $A \land (A \lor X) = A$ ):

A \lor (B \land (A \lor \neg B))

Step 3: Simplify $(B \land (A \lor \neg B))$ using Distribution:

A \lor (B \land A) \lor (B \land \neg B)

Step 4: Apply Identity $(B \land \neg B = 0)$

A \lor (B \land A) \lor 0

Step 5: Simplify further:

Result = A

Note Summary

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Common Mistakes

Misapplying De Morgan's Laws: Students often forget to flip the logical operator when applying these laws.

For example: Incorrect: $\neg(A \land B) = \neg A \land \neg B$

Skipping Steps in Distribution: When distributing, ensure every term is expanded properly.

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For example: $A \lor (B \land C)$ should become $(A \lor B) \land (A \lor C)$ , not just $A \lor B \land C$

Confusing Associative and Commutative Properties: Associative refers to grouping (changing parentheses), while Commutative refers to order (swapping terms). These rules apply to both AND and OR but not negations.

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Key Takeaways

De Morgan's Laws help transform negated AND/OR operations.
Distribution allows logical expressions to be factored or expanded.
Association and Commutation enable expressions to be rearranged without affecting results.
Double Negation simplifies two negations to the original term.
Always simplify step-by-step to avoid errors and ensure clarity.

Simplifying Boolean Algebra Simplified Revision Notes for A-Level OCR Computer Science

Simplifying Boolean Algebra

Overview

De Morgan's Laws

Distribution

Distribute OR over AND:

Distribute AND over OR:

Association

Commutation

Double Negation

Examples

Note Summary

Common Mistakes

Key Takeaways

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