Simplifying Boolean Expressions (De Morgan's Laws)

Boolean expressions can often be simplified to use fewer gates, which makes circuits cheaper, faster, and more energy-efficient. OCR J277 Section 2.5 expects you to be able to simplify simple Boolean expressions and understand De Morgan's Laws.

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Why Simplify?

Unsimplified	Simplified	Benefit
More gates needed	Fewer gates needed	Lower cost
More connections	Fewer connections	Less power consumption
Longer signal paths	Shorter signal paths	Faster processing
More potential faults	Fewer potential faults	Higher reliability

Simplification does not change what the circuit does — it produces an equivalent expression that gives the same output for every possible input combination.

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Basic Boolean Algebra Laws

Before tackling De Morgan's Laws, you should know these fundamental identities:

Identity Laws

• A AND 1 = A (ANDing with 1 has no effect)
• A OR 0 = A (ORing with 0 has no effect)

Null Laws

• A AND 0 = 0 (ANDing with 0 always gives 0)
• A OR 1 = 1 (ORing with 1 always gives 1)

Complement Laws

• A AND (NOT A) = 0 (a value ANDed with its inverse is always 0)
• A OR (NOT A) = 1 (a value ORed with its inverse is always 1)

Idempotent Laws

• A AND A = A
• A OR A = A

Double Negation

• NOT (NOT A) = A (inverting twice returns to the original)

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Simplification Examples

Example 1: A AND (A OR B)

Using the absorption law: A AND (A OR B) = A

Proof by truth table:

A	B	A OR B	A AND (A OR B)	A
0	0	0	0	0
0	1	1	0	0
1	0	1	1	1
1	1	1	1	1

The last two columns are identical, confirming A AND (A OR B) = A.

Example 2: (A AND B) OR (A AND (NOT B))

Factor out A: A AND (B OR (NOT B)) = A AND 1 = A

This uses the complement law: B OR (NOT B) = 1, and the identity law: A AND 1 = A.

OCR Exam Tip: When simplifying, look for variables that appear both as themselves and as their NOT form — you can often use the complement law to simplify them to 1 or 0.

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De Morgan's Laws

Augustus De Morgan (1806-1871) discovered two laws that show how to convert between AND and OR operations when combined with NOT:

First Law

NOT (A AND B) = (NOT A) OR (NOT B)