Boolean algebra and De Morgan's laws
Boolean algebra lets you simplify a logic expression before building it, which means fewer gates. A dozen identities and De Morgan's laws are enough for every exam question; the skill is choosing which to apply next.
Part 1 of 3: Learn it
In short
- Here A.B means A AND B, A + B means A OR B, and ¬A means NOT A.
- De Morgan: ¬(A.B) = ¬A + ¬B and ¬(A + B) = ¬A.¬B.
- Absorption: A + A.B = A and A.(A + B) = A.
Where this is in your specification
Spec points: AQA 7517 4.6.4 and 4.6.5, OCR H446 1.4.3
The identities
| AND form | OR form |
|---|---|
| A.0 = 0 | A + 1 = 1 |
| A.1 = A | A + 0 = A |
| A.A = A | A + A = A |
| A.¬A = 0 | A + ¬A = 1 |
| ¬¬A = A | |
| A.(B + C) = A.B + A.C | A + B.C = (A + B).(A + C) |
Commutative and associative rules also hold: the order of terms and the grouping of a run of ANDs (or of ORs) does not matter.
De Morgan's laws
To apply one: break the bar over the whole bracket, change the operator inside (AND to OR, or OR to AND), and put a NOT on each term. It works the other way too, which is how a long expression is rewritten to use only NAND or only NOR gates.
Universal gates
NAND on its own can make every other gate, and so can NOR. For example, NOT A is A NAND A, and A AND B is (A NAND B) NAND (A NAND B). Chips are often built from one gate type because it is cheaper to manufacture.
Simplify A + A.B.
Show the answer
A (absorption).
Part 2 of 3: See it worked
Worked examples
Example 1
Simplify A.B + A.¬B.
- Factorise: A.(B + ¬B)
- B + ¬B = 1
- A.1 = A
Answer: A.
Example 2
Simplify ¬(¬A.B).
- De Morgan: ¬¬A + ¬B
- Double negation: ¬¬A = A
Answer: A + ¬B.
Common mistakes
- Applying De Morgan without changing the operator.
- Changing the operator but forgetting to negate each term.
- Writing A + A = 2A. In Boolean algebra A + A = A.
- Skipping steps so that a method mark cannot be given.
Use De Morgan's law to rewrite ¬(A + B).
Show the answer
¬A.¬B.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Simplify A + A.B.
A (absorption).
Use De Morgan's law to rewrite ¬(A + B).
¬A.¬B.
How can a NOT gate be made from a NAND gate?
Connect both inputs of the NAND gate to A.
Jobs that use this
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses