Boolean Algebra: Duality & Identities

Boolean equations maintain validity under duality when both sides' expressions are interchanged.
Interchanging logical operators and replacing constants: OR ( + ) with AND ( * ) and 1's with 0's yields the dual expression.

Identity with 0: x + 0 = x; acts as an identity element for OR operation.
Identity with 1: x + 1 = 1; represents the absorbing element for OR operation.
Idempotent Law: x + x = x; combining the same variable does not change the outcome.
Complements: x + x' = 1; a variable ORed with its complement equals 1.
Double Negation: (x')' = x; negating a variable twice returns the original variable.
Identity for AND: x(1) = x; 1 acts as an identity element for AND operation.
Null Law: x(0) = 0; combining any variable with 0 results in 0.
Idempotent for AND: x * x = x; multiplying the same variable yields the same variable.
Complement Law: x(x') = 0; a variable ANDed with its complement equals 0.

Commutative Property: Order of operations does not affect outcome; x + y = y + x and x * y = y * x.
Associative Property: Grouping of variables does not affect outcome; x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z.
Distributive Property: Describes how operations interact; x * (y + z) = xy + xz, defining distribution of AND over OR.

Transformation rules for negation: (x + y)' = x' * y' and (x * y)' = x' + y'; critical for simplifying expressions.
Useful for converting between AND/OR and their complements, aiding in circuit design and logical proofs.