Cambridge IGCSE Computer Science (0478) · lesson 35 of 35 · 30 min
Boolean Algebra Laws
De Morgan’s Theorems
De Morgan’s theorems allow you to convert between AND and OR forms:
NOT(A AND B) = NOT A OR NOT B
NOT(A OR B) = NOT A AND NOT B
These are extremely useful for simplifying Boolean expressions and designing equivalent circuits with fewer gates.
Worked Example — Simplification
Simplify: NOT(A AND B) AND (A OR B)
Step 1: Apply De Morgan’s: NOT(A AND B) = NOT A OR NOT B
Expression becomes: (NOT A OR NOT B) AND (A OR B)
Step 2: Expand using distributive law:
(NOT A AND A) OR (NOT A AND B) OR (NOT B AND A) OR (NOT B AND B)
Step 3: Apply complement law: NOT A AND A = 0, NOT B AND B = 0
= 0 OR (NOT A AND B) OR (A AND NOT B) OR 0
Step 4: Apply identity law (OR 0):
= (NOT A AND B) OR (A AND NOT B) = A XOR B
According to De Morgan’s theorem, NOT(A OR B) equals:
- NOT A OR NOT B
- NOT A AND NOT B
- A AND B
- NOT A OR B
- Answer: 1
De Morgan: NOT(A OR B) = NOT A AND NOT B.
Simplify: A AND 1
- 0
- 1
- A
- NOT A
Answer: A
Identity law: A AND 1 = A.
What does A OR NOT A always equal?
- 0
- A
- 1
- NOT A
Answer: 1
Complement law: A OR NOT A = 1 (always true).
Previous lesson: Logic gates and truth tables
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