l3_15a_6

# l3_15a_6 - Last time: Fundamental laws of the algebra of...

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1 ECE 15A Fundamentals of Logic Design Lecture 3 Malgorzata Marek-Sadowska Electrical and Computer Engineering Department UCSB 2 Last time: Fundamental laws of the algebra of sets Comutative laws (1a) XY = YX (1b) X+Y = Y+X Associative laws (2a) X(YZ) = (XY)Z (2b) X+(Y+Z) = (X+Y)+Z Distributive laws (3a) X(Y+Z) = XY+XZ (3b) X+YZ= (X+Y)(X+Z) Laws of Tautology (4a) XX = X (4b) X+X = X Laws of Absorbtion (5a) X(X+Y) = X (5b) X+XY = X 3 Last time: Fundamental laws of the algebra of sets Laws of Complementation (6a) XX’=0 (6b) X+X’=1 Double Laws of Complementation (7) (X’)’=X Laws of De Morgan (8a) (XY)’ = X’+Y’ (8b) (X+Y)’ = X’Y’ Operations with 0 and 1 (9a) 0X=0 (9b) 1+X=1 (10a) 1X=X (10b)0+X=X (11a) 0’=1 (11b) 1’=0 4 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. 5 Today: ± General treatment of Boolean algebras based on definitions and axioms ² Some duplication of the material introduced in lecture #2 ² Earlier results were obtained based on intuitive concepts 6 Sets with algebraic structure ± Rules of combination defined between elements of the set ± Examples: ² Set of all integers ² Set of all real numbers ¾ Rules of combination: addition, subtraction, multiplication, division ² Sets ¾ Intersection and union

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2 7 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 S. We write c=a & b. ± Examples ² Subtraction is a binary operation on the set or rational numbers ² Subtraction is not a binary operation on the set of natural numbers ² Subtraction is a binary operation on the set of all integers. ± Binary operation doesn’t need to have an intuitive meaning (like +, -, /, x). 8 Examples of abstract binary operations ± Let S= { A, B, C }. ± We define # and & as follows: # 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 A & B = B B & A = A 9 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) 10 Example ± Algebra of sets ² Intersection and union are commutative and associative ² Each is distributive over the other
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## l3_15a_6 - Last time: Fundamental laws of the algebra of...

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