Boolean Algebra

Negation Laws

   ¬(¬a) = a
   ¬a ∨ a = true
   ¬a ∧ a = false
   a ∨ ¬a ∧ b = a ∨ b

AND Identify Laws

   false ∧ a = false
   true ∧ a = a
   a ∧ a = a
   a ∧ ¬a = false

OR Identify Laws

   false ∨ a = a
   true ∨ a = true
   a ∨ a = a
   a ∨ ¬a = true

XOR Identify Laws

   false ⊕ a = a
   true ⊕ a = ¬a
   a ⊕ a = false
   a ⊕ ¬a = true

XOR Equivalence Laws

   a ⊕ b = (a ∧ ¬b) ∨ (¬a ∧ b)
   a ⊕ b = (a ∨ b) ∧ (¬a ∨ ¬b)
   a ⊕ a = (a ∨ b) ∧ ¬(a ∧ b)

Commutativity Laws

   a ∨ b = b ∨ a
   a ∧ b = b ∧ a
   a ⊕ b = b ⊕ a

Associativity Laws

   (a ∨ b) ∨ c = a ∨ (b ∨ c)
   (a ∧ b) ∧ c = a ∧ (b ∧ c)
   (a ⊕ b) ⊕ c = a ⊕ (b ⊕ c)

Implication Laws

   a → b = ¬a ∨ b
   a → b = ¬b → ¬a
   true → a = a
   false → a = true

Distributive Laws

   a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
   a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

DeMorgan's Laws

   ¬(a ∨ b) = ¬a ∧ ¬b
   ¬(a ∧ b) = ¬a ∨ ¬b


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Last updated 2025-10-07 21:41