Boolean algebra

Basic operations

Has only three basic operations: conjunction, disjunction, and negation, expressed with the corresponding binary operators AND ( ∧ or ⋅ ) and OR ( ∨ or + ) and the unary operator NOT ( ¬ or ∼ )

Secondary operations

Material conditional: x → y = ¬ x ∨ y Material biconditional: x ↔ y = ( x ∧ y ) ∨ ( ¬ x ∧ ¬ y ) = ( x ∨ ¬ y ) ∧ ( ¬ x ∨ y ) Exclusive OR (XOR): x ⊕ y = ¬ ( x ↔ y ) = ( x ∨ y ) ∧ ¬ ( x ∧ y ) = ( x ∨ y ) ∧ ( ¬ x ∨ ¬ y ) = ( x ∧ ¬ y ) ∨ ( ¬ x ∧ y )

XOR Operation ⊕ Returns true if exactly one input is true. NAND Operation ↓ Returns false only if both inputs are true. NOR Operation ↑ Returns false if at least one input is true. XNOR Operation ↔ Returns true if both inputs are equal.