Chapter_2_Boolean_Algebra_and_Logic_Gates

# Chapter_2_Boolean_Algebra_and_Logic_Gates - Digital Logic...

This preview shows pages 1–7. Sign up to view the full content.

May 26, 2011 EASTERN MEDITERRANEAN UNIVERSITY 1 Digital Logic Design I Boolean Algebra and Logic Gate Mustafa Kemal Uyguroğlu

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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. 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 )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern