# 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