ece15_3_2011_6

ece15_3_2011_6 - Boolean Algebra The 19th Century...

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1 ECE 15A Fundamentals of Logic Design Lecture 3 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Boolean Algebra The 19th Century Mathematician, George Boole, developed a math. system (algebra) involving formal principles of reasoning, Boolean Algebra. Later Claude Shannon (father of information theory) showed (in his Master’s thesis!) how to map Boolean Algebra to digital circuits. 3 Today: General treatment of Boolean algebras based on definitions and axioms Some duplication of the material introduced in lecture #2 Earlier results were obtained based on intuitive concepts 4 Sets with algebraic structure Rules of combination defined between elements of the set Examples: Set of all integers Set of all real numbers Rules of combination: addition, subtraction, multiplication, division Sets Intersection and union 5 Binary operation A binary operation “&” on a set S is a rule which for each ordered pair of elements (a,b) s.t. a,b belong to S, assigns a unique element c Examples Subtraction is a binary operation on the set or rational numbers Subtraction is not a binary operation on the set of natural numbers Subtraction is a binary operation on the set of all integers. Binary operation doesn’t need to have an intuitive meaning (like +, -, /, x). 6 Examples of abstract binary operations Let S= { A, B, C }. # A B C A A C B B C B A C B A C & A B C A A B C B A B C C A B C A # B = C B # C = A B & A = A
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2 7 Definitions A binary operation @ on a set of elements S is associative if and only if for every a,b,c S a @ (b @ c) = (a @ b) @ c A binary operation @ on a set of elements S is commutative if and only if for every a,b S a @ b = b @ a If @ and % are two binary operations on the same set S, @ is distributive over % if and only if for every a,b,c S a @ (b % c) = (a @ b) % (a @ c) 8 Example Let S= { A, B, C }. # A B C A A C B B C B A C B A C & A B C A A B C B A B C C A B C Is # associative? (A#B)#C = C#C= C A #(B#C) = A#A = A Is & associative? 9 Example Algebra of sets Intersection and union are commutative and associative Each is distributive over the other X(Y+Z) = XY+XZ X+YZ= (X+Y)(X+Z) 10 Example Natural numbers Addition and multiplication are commutative and associative Multiplication is distributive over addition Addition is not distributive over multiplication
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ece15_3_2011_6 - Boolean Algebra The 19th Century...

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