De Morgan's laws are two transformation rules that let you rewrite any Boolean expression involving NOT with AND or OR. The first law states that NOT(A AND B) equals NOT A OR NOT B; the second states that NOT(A OR B) equals NOT A AND NOT B. These laws are essential for simplifying logic gate circuits at GCSE.
Who was Augustus De Morgan?
Augustus De Morgan was a nineteenth-century British mathematician who formalised the relationship between AND and OR operations when combined with NOT. His laws, published in 1847, showed that negating a compound Boolean expression is equivalent to negating each individual term and switching the operator. Turing, working a century later, built on exactly this kind of Boolean algebra to design the logical structure of computing machines.
The laws appear in both symbolic mathematics and digital electronics — any circuit designer who wants to replace an AND gate and a NOT gate with a NAND gate is applying De Morgan's laws.
What are De Morgan's two laws?
First law (NOT AND → OR NOT):
NOT(A AND B) = NOT A OR NOT B
In notation you may see in your specification:
¬(A ∧ B) = ¬A ∨ ¬B
or equivalently: A NAND B = (NOT A) OR (NOT B)
Second law (NOT OR → AND NOT):
NOT(A OR B) = NOT A AND NOT B
¬(A ∨ B) = ¬A ∧ ¬B
or equivalently: A NOR B = (NOT A) AND (NOT B)
A useful way to remember both laws:
- Break the bar, change the operator. When you distribute a NOT across a bracket: NOT each term, and flip AND↔OR.
Proving the first law with a truth table
NOT(A AND B) = NOT A OR NOT B
| A | B | A AND B | NOT(A AND B) | NOT A | NOT B | NOT A OR NOT B |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 | 0 | 1 | 1 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
The columns NOT(A AND B) and NOT A OR NOT B are identical for every combination of A and B. The two expressions are logically equivalent. ✓
Proving the second law with a truth table
NOT(A OR B) = NOT A AND NOT B
| A | B | A OR B | NOT(A OR B) | NOT A | NOT B | NOT A AND NOT B |
|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 1 | 1 | 1 |
| 0 | 1 | 1 | 0 | 1 | 0 | 0 |
| 1 | 0 | 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 0 | 0 | 0 | 0 |
Again, the two final columns are identical. ✓
How do you apply De Morgan's laws to simplify an expression?
Worked example 1:
Simplify: NOT(P AND Q) AND R
Step 1 — Apply the first law to NOT(P AND Q):
NOT(P AND Q) = NOT P OR NOT Q
Step 2 — Substitute:
(NOT P OR NOT Q) AND R
This cannot be simplified further without additional information. But the equivalent circuit has changed: instead of a NOT gate fed by an AND gate, you now have an OR gate fed by two separate NOT gates — which may use fewer physical components in a circuit.
Worked example 2:
Simplify: NOT(X OR NOT Y)
Step 1 — Apply the second law to NOT(X OR NOT Y):
NOT X AND NOT(NOT Y)
Step 2 — NOT(NOT Y) = Y (double negation cancels):
NOT X AND Y
This is a significant simplification — three operations collapsed to two.
Why are De Morgan's laws useful in circuit design?
NAND and NOR gates are physically simpler and cheaper to manufacture than AND and OR gates in CMOS transistor technology. De Morgan's laws allow engineers to replace combinations of AND/OR/NOT gates with equivalent NAND or NOR gates, reducing component count and power consumption.
| Original expression | De Morgan equivalent | Equivalent gate |
|---|---|---|
| NOT(A AND B) | NOT A OR NOT B | NAND gate |
| NOT(A OR B) | NOT A AND NOT B | NOR gate |
At GCSE, you may be asked to redraw a circuit using only NAND gates or only NOR gates. De Morgan's laws are the tool that makes this transformation possible.
How are De Morgan's laws tested in GCSE exams?
Exam questions typically fall into these categories:
- Complete a truth table for an expression such as NOT(A AND B) and a second expression such as NOT A OR NOT B, then identify whether they are equivalent.
- Simplify a Boolean expression step by step, citing the law used at each step.
- Rewrite a circuit using an equivalent arrangement of gates, applying the law to replace a NOT-AND or NOT-OR combination.
- Describe in words what De Morgan's first/second law states, without symbols.
For all question types, the two tables above (truth table proofs) are the most reliable check: if two expressions produce identical output columns, they are equivalent.
Frequently asked questions
Do I need to use formal symbols (¬, ∧, ∨) or can I write NOT, AND, OR?
Both notations are accepted in GCSE exams, but check what your exam board uses in its mark schemes. AQA tends to use NOT, AND, OR in words or dot/plus notation (A·B, A+B, Ā). OCR similarly uses text and overbar notation. Use whichever your teacher has taught you and be consistent within an answer.
What is the difference between De Morgan's laws and Boolean identities such as A AND 0 = 0?
Boolean identities (also called Boolean axioms or laws of Boolean algebra) include simplification rules such as A AND 0 = 0, A AND 1 = A, A OR A = A, and A AND NOT A = 0. De Morgan's laws are specifically about distributing NOT across a compound AND or OR expression. Both types of rule are used together when fully simplifying a complex Boolean expression.
Is it possible to have a double NOT cancel out?
Yes. NOT(NOT A) = A always — applying NOT twice returns to the original value. This is called the double negation law. It is useful when applying De Morgan's laws introduces NOT(NOT Y) terms, as shown in Worked Example 2 above.
Are De Morgan's laws on the AQA GCSE specification?
De Morgan's laws are explicitly named in the OCR GCSE Computer Science specification (J277). AQA GCSE Computer Science covers Boolean algebra and logic simplification, and De Morgan's laws form part of that content — they are used when simplifying expressions and when completing truth tables for NAND and NOR gates. Check the specific topic list in your exam board's specification document to confirm the exact scope for your course.
Work through Boolean algebra and logic gate simplification with Professor Turing's guided hints at aitutors.me.