Chapter_2_Boolean_Algebra_and_Logic_Gates

Chapter_2_Boolean_Algebra_and_Logic_Gates - EASTERN...

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Unformatted text preview: May 26, 2011 EASTERN MEDITERRANEAN UNIVERSITY 1 Digital Logic Design I Boolean Algebra and Logic Gate Mustafa Kemal Uygurolu May 26, 2011 2 Algebras What is an algebra? Mathematical system consisting of Set of elements Set of operators Axioms or postulates Why is it important? Defines rules of calculations Example: arithmetic on natural numbers Set of elements: N = {1,2,3,4,} Operator: +, , * Axioms: associativity, distributivity, closure, identity elements, etc. Note: operators with two inputs are called binary Does not mean they are restricted to binary numbers! Operator(s) with one input are called unary May 26, 2011 3 BASIC DEFINITIONS A set is collection of having the same property. S : set, x and y : element or event For example: S = {1, 2, 3, 4} If x = 2, then x S . If y = 5, then y S . A binary operator defines on a set S of elements is a rule that assigns, to each pair of elements from S, a unique element from S. For example: given a set S , consider a * b = c and * is a binary operator. If ( a , b ) through * get c and a , b , c S , then * is a binary operator of S . On the other hand, if * is not a binary operator of S and a , b S , then c S . May 26, 2011 4 BASIC DEFINITIONS The most common postulates used to formulate various algebraic structures are as follows: 1. Closure : a set S is closed with respect to a binary operator if, for every pair of elements of S , the binary operator specifies a rule for obtaining a unique element of S . For example, natural numbers N={1,2,3,...} is closed w.r.t. the binary operator + by the rule of arithmetic addition, since, for any a , b N, there is a unique c N such that a+b = c But operator is not closed for N , because 2-3 = -1 and 2, 3 N , but (-1) N. 2. Associative law : a binary operator * on a set S is said to be associative whenever ( x * y ) * z = x * ( y * z ) for all x , y , z S ( x+y ) +z = x+ ( y+z ) 3. Commutative law : a binary operator * on a set S is said to be commutative whenever x * y = y * x for all x , y S x+y = y+x May 26, 2011 5 BASIC DEFINITIONS 1. Identity element : a set S is said to have an identity element with respect to a binary operation * on S if there exists an element e S with the property that e * x = x * e = x for every x S 0+x = x+0 =x for every x I . I = {, -3, -2, -1, 0, 1, 2, 3, }. 1*x = x*1 =x for every x I. I = {, -3, -2, -1, 0, 1, 2, 3, }. 2. Inverse : a set having the identity element e with respect to the binary operator to have an inverse whenever, for every x S , there exists an element y S such that x * y = e The operator + over I , with e = 0, the inverse of an element a is (- a ), since a +(- a ) = 0....
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This note was uploaded on 05/26/2011 for the course EE 211 taught by Professor Hasandemirel during the Spring '10 term at Eastern Mediterranean University.

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Chapter_2_Boolean_Algebra_and_Logic_Gates - EASTERN...

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