Boolean Algebra Laws

Boolean Algebra: axioms, laws, identities, principles

law	AND	OR
identity	a * 1 = a	a + 0 = a
identity element	1	0
domination	a * 0 = 0	a + 1 = 1
zero element	0	1

idempotence | a * a = a | a + a = a inverse | a * a' = 0 | a + a' = 1 commutativity | ab = ba | a+b = b+a associativity | (ab)c = a(bc) | (a + b) + c = a + (b + c) distributivity | (a+b)c = (a+b)(a+c) | a * (b + c) = a * b + a * c absorption | a * (a+b) = a | a + a * b = a de morgan's law | (ab)' = a' + b' | (a + b)' = a' * b' double complement | a'' = a | a'' = a

Laws (⊛ stands for both ops):

• Dominance/null law: a + 1 = 1, a * 0 = 0 a ⊛ x = x

• Inverse law: a + a' = 1, a * a' = 0

• Double Complement Law: a <=> a''

• Commutative law: a + b <=> b + a, ab <=> ba

• Associative law: (a ⊛ b) ⊛ c <=> a ⊛ (b ⊛ c)

• De Morgan's Law: (a * b)' <=> a' + b', (a + b)' <=> a' * b'

• Distributive law: a * (b + b') <=> a * b + a * b'

• Absorption law: a + (a * b) = a, a * (a + b) = a, b'+(a b')=b' Like terms are absorbed. Opposite operators must be used within and outside the brackets, for absorption to be used. The term that is outside parens must also be inside.

• ⊛ - when only operator, stands for both ops

• ⊛, ★ - the ⊛ stands for one operation, ★ for the other

(a ⊛ b)' <=> a' ★ b' ¬(a ⊛ b) <=> ¬a ★ ¬b (a * b)' <=> a' + b' ¬(a ∧ b) <=> ¬a ∨ ¬b (a + b)' <=> a' * b' ¬(a ∨ b) <=> ¬a ∧ ¬b