# Basic Theorems and Properties of Boolean Algebra

May 29th, 2015
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Duality - We simply interchange OR and AND operators and replace 1’s by 0’s and 0’s by 1’s. Also, the duality principle states that if two Boolean expressions are equal, then their duals are also equal. Ex: X + 0 = X & X •1 = X

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Chapter 2Boolean Algebra Logic and Logic GatesBasic Theorems and Properties of Boolean Algebra:DualityWe simply interchange OR and AND operators and replace 1s by 0s and 0s by 1s. Also, the duality principle states that if two Boolean expressions are equal, then their duals are also equal. Ex: X + 0 = X & X 1 = XBasic TheoremBoolean Addition:x + 0 = 0 + x = x (0 is called the additive identity)x + 1 = 1 (X = AB + C AB + C + 1 = 1)Boolean Multiplicationx 0 = 0x 1 = 1 x = x (1 is called the multiplicative identity)Idempotent Lawsx + x = xx x = xInvolution Law(x')' = xLaws of Complementx + x' = 1x x' = 0Commutative Lawsx + y = y + xx y = y xAssociative Laws(x + y) + z = x + (y + z) = x + y + z(x y) z = x (y z) = x y zDistributive Lawx (y + z) = (x y) + (x z)x + (y z) = (x + y) (x + z)DeMorgan's Laws(x + y)' = x' y'(x y)' = x' + y'Consensus Lawx y + x' z + y z = x y + x' zThe term yz is referred to as the "consensus term". A consensus term is a redundant term and it can be eliminated. Given a pair of terms for which a variable appears in one term and the complement of that variable in another term, the consensus term is formed by multiplying the two original terms together, leaving out the selected variables and its complement.A field is a set of elements, together with two binary operators.The set of real numbers together with the binary operators + and

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