De Morgan's Laws, distribution, association, commutation and double negation

Use the following rules to derive or simplify statements in Boolean algebra: De Morgan’s Laws, distribution, association, commutation, double negation

1.4.3 Boolean Algebra

Boolean Algebra — Note 3 of 5

Candidates should be aware of the given De Morgan’s laws and should be able to apply these to a Boolean statement. Candidates should have experience of manipulating and simplifying Boolean statements using these rules of distribution, commutation, association and double negation.

What you need to know

Boolean algebra has rules that let expressions be rearranged and simplified. You need to apply De Morgan's laws, distribution, association, commutation and double negation to derive or simplify Boolean statements.

What to know

De Morgan's laws: ¬(A ∧ B) ≡ ¬A ∨ ¬B and ¬(A ∨ B) ≡ ¬A ∧ ¬B. To apply: change the operator, then negate each term.
Distribution: A ∧ (B ∨ C) ≡ (A ∧ B) ∨ (A ∧ C) and A ∨ (B ∧ C) ≡ (A ∨ B) ∧ (A ∨ C).
Association: (A ∧ B) ∧ C ≡ A ∧ (B ∧ C) and (A ∨ B) ∨ C ≡ A ∨ (B ∨ C).
Commutation: A ∧ B ≡ B ∧ A and A ∨ B ≡ B ∨ A.
Double negation: ¬¬A ≡ A.
Other useful identities: A ∧ ¬A = 0, A ∨ ¬A = 1, A ∧ A = A, A ∨ 1 = 1, A ∧ 0 = 0, A ∨ (A ∧ B) ≡ A (absorption).

Exam technique

Tip:Write each step on a new line and name the rule you used; method marks are available for correct steps.
Mistake:Applying De Morgan's law without changing the operator (e.g. writing ¬(A ∧ B) ≡ ¬A ∧ ¬B).

Code examples

Worked simplification

text

¬(¬A ∨ B)
≡ ¬¬A ∧ ¬B      De Morgan's law
≡ A ∧ ¬B        Double negation