Boolean Algebra (Cambridge International A-Level Computer Science)

Exam board Cambridge International · A-LevelUpdated 2026-10-01

Boolean algebra

What is Boolean algebra?

Boolean algebra is a mathematical system used to manipulate Boolean values
Complex expressions can be made simpler using the rules of Boolean algebra
This is a more powerful simplification method than Karnaugh maps and can simplify expressions that Karnaugh maps cannot
There are various different **rules **that you need to learn and that can then be applied to certain expressions to simplify them
Combining these rules can mean that complex expressions can be** reduced to much simpler ones**

General rules

General AND rules
- X AND 0 = 0
- X AND 1 = X
- X AND X = X
- NOT X AND X = 0
Note, the value of X is unknown and it is used as a placeholder
Therefore X AND 1 = X means that the output will be whatever the value of X is
General OR rules
- X OR 0 = X X OR 1 = 1 X OR X = X NOT X OR X = 1

Boolean algebra notation

In Boolean algebra, expressions are written using shorthand notation:

Symbol	Meaning	Example	Explanation
$A \times B$ or `AB`	AND	`A · B`	True only if both A and B are 1
$A+B$	OR	`A + B`	True if either A or B is 1
$\overline{A}$	NOT / complement	$\overline{A}$	True if A is 0, False if A is 1
`( )`	Brackets / grouping	`(A + B)·C`	Do A OR B first, then AND with C

Exam Hint A dot (·) for AND is often omitted, so AB means A AND B

A line drawn above a variable or expression means that the value is inverted or negated
It's the NOT of that value

Notation	Meaning	Explanation
$\overline{A}$	NOT A	True if A is False, False if A is True
$A.\overline{B}$	A AND (NOT B)	B is negated before the AND operation
$\overline{A+B}$	NOT (A OR B)	The entire OR expression is negated (use De Morgan)
$\overline{A.B.C}$	NOT (A AND B AND C)	All values are ANDed together, then the result is negated

When you see multiple horizontal lines, for example:

Three Boolean expressions: (A.B), (A.C), (B.D) each with an overline indicating negation, displayed as digital logic notation.

It means the whole expression is negated, not just one part
Always apply De Morgan’s Law starting with the outermost line first

De Morgan's Law

What is De Morgan’s Law?

De Morgan’s Laws are used to simplify Boolean expressions involving negation of conjunctions (AND) or disjunctions (OR)
They are particularly useful for rewriting logic circuits using only NAND or NOR gates
There are two key rules:
1. NOT (A AND B) is equivalent to (NOT A) OR (NOT B) $\overline{A.B}=\overline{A}+\overline{B}$
2. NOT (A OR B) is equivalent to (NOT A) AND (NOT B) A + B -bar = top A . top B

Worked example 1: Simplify ¬(A ∧ B)

Step	Explanation	Expression
Step 1	Start with the original expression	`top A . B`
Step 2	Apply De Morgan’s Law: change ∧ to ∨ and negate both terms	$\vec{A} + \vec{B}$
Step 3	Remove unnecessary brackets	$\vec{A} + \vec{B}$

Worked example 2: Simplify ¬(A ∨ B)

Step	Explanation	Expression
Step 1	Start with the original expression	`top A + B`
Step 2	Apply De Morgan’s Law: change ∨ to ∧ and negate both terms	$\vec{A} . \vec{B}$
Step 3	Remove unnecessary brackets	$\vec{A} . \vec{B}$

Why use De Morgan’s Laws?

Simplifying Boolean expressions using De Morgan’s Laws:
- Allows for implementation using only NAND or NOR gates
- Makes circuit design more efficient and cost-effective
- Is essential in hardware where NAND/NOR logic is cheaper or faster to implement
- Is commonly used in microprocessor and memory device design (e.g. flash drives)

Worked Example Simplify the following expression using Boolean algebra, including De Morgan’s laws. Show your working.[3]

Final answer: Answer

Correct application of De Morgan’s Law [1 mark]

Correct application of Double Negation Law or Distributive Law [1 mark]

Correct answer [1 mark]

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Symbol

Meaning

Example

Explanation

A \times B

or AB

AND

A · B

True only if both A and B are 1

A+B

A + B

True if either A or B is 1

\overline{A}

NOT / complement

\overline{A}

True if A is 0, False if A is 1

( )

Brackets / grouping

(A + B)·C

Do A OR B first, then AND with C

Notation

Meaning

Explanation

\overline{A}

NOT A

True if A is False, False if A is True

A.\overline{B}

A AND (NOT B)

B is negated before the AND operation

\overline{A+B}

NOT (A OR B)

The entire OR expression is negated (use De Morgan)

\overline{A.B.C}

NOT (A AND B AND C)

All values are ANDed together, then the result is negated

Step

Explanation

Expression

Step 1

Start with the original expression

top A . B

Step 2

Apply De Morgan’s Law: change ∧ to ∨ and negate both terms

\vec{A} + \vec{B}

Step 3

Remove unnecessary brackets

\vec{A} + \vec{B}

Step

Explanation

Expression

Step 1

Start with the original expression

top A + B

Step 2

Apply De Morgan’s Law: change ∨ to ∧ and negate both terms

\vec{A} . \vec{B}

Step 3

Remove unnecessary brackets

\vec{A} . \vec{B}