Chapter_2_Boolean_Algebra_and_Logic_Gates

Chapter_2_Boolean_Algebra_and_Logic_Gates - Digital Logic...

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May 26, 2011 EASTERN MEDITERRANEAN UNIVERSITY 1 Digital Logic Design I Boolean Algebra and Logic Gate Mustafa Kemal Uyguroğlu
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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
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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 .
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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
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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. 3. Distributive law : if * and u are two binary operators on a set S, * is said to be distributive over . whenever x * ( y b z ) = ( x * y ) u ( x * z )
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May 26, 2011 6 George Boole Father of Boolean algebra He came up with a type of linguistic algebra, the three most basic operations of which were (and still are) AND, OR and NOT. It was these three functions that formed the basis of his premise, and were the only operations
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