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<p>In other words, we show  Boolean Algebra. Boolean algebra. a + a' =a · a' =Order of Evaluation of Boolean Expressions A Boolean algebrais a set Bof values together withtwo binary operations, commonly denoted by + and ∙,a unary operation, usually denoted by ˉ or ~ or ’,two elements  Useful laws and theorems. Axioms  TheoremDe Morgan’s Law. Theorem: For every pair a, b in set B: (a+b)’ = a’b’, and (ab)’ = a’+b’. Proof: We show that a+b and a’b’ are complementary. Boolean algebra is a deductive mathematical system closed over the values zero and one (false and true). A Boolean algebrais a set Bof values together withtwo binary operations, commonly denoted by + and ∙,a unary operation, usually denoted by ˉ or ~ or ’,two elements usually called zeroand one, such that for every element xof B: In addition, certain axioms must be satisfied CNF: Complete OperationA B C OUT Expressionnot ((not A) and (not B) and (not C))not (A and (not B) and (not C))not (A and B and Duality (a meta-theorem— a theorem about theorems) NullAll Boolean expressions have logical duals Any theorem that can be proved is also proved for its dual Replace: with +, + with •,with 1, andwithLeave the variables unchanged Example: The dual of X+0= X is X•1= XUseful laws and theorems Identity X+0 = X Dual: X•1 = X There exists a basic duality which underlies all Boolean algebra. For every a, b, and c in B, a + (b · c) = (a + b) · (a + c) a · (b + c) = (a · b) + (a · c) Axiom– Complement. A. binary operator. A literal represents the connection of a variable or its complement to a unique gate input. B. binary operators (+, •) unary operator (' or) Binary logic. The laws and theorems which have been presented can all be divided into pairs. In proving the laws and theorems, it is then necessary only to prove one theorem, and the “dual” of the theorem follows necessarily. Do not include the output variable. •If Eand Eare two expressions for the same Boolean function (i.e., they CS@VT© WD McQuain. Examples. a ·= a, for every a in B. Axioms of Boolean Algebra (4 of 4) Axiom– Distributive laws. To form the dual of an algebraic expression you simply need to Boolean Algebra. For each a in B, there exists an element a' in B (the complement of a) s.t. A Boolean algebra consists of a set of elements. ” deﬁned over this set of •In a Boolean expression, each variable’s appearance in either its non-complemented or complemented form is called a literal.</p></translate>
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