Chapter 4 Boolean Algebra PDF

4-1 Computer Architecture and Organization Chapter 4 – Boolean Algebra 4-2 Learning Outcomes o To describe the fundamental concept of Boolean algebra, Boolean functions, truth tables, algebra manipulation, Boolean expressions and simplification of Boolean Fu...

4-1 Computer Architecture and Organization Chapter 4 – Boolean Algebra 4-2 Learning Outcomes o To describe the fundamental concept of Boolean algebra, Boolean functions, truth tables, algebra manipulation, Boolean expressions and simplification of Boolean Functions o To explain the correspondence between electronic circuits and boolean functions, combinational logic and sequential Logic o To draw the truth table and circuit diagram. 4-3 Concept of Boolean algebra  Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0 respectively.  Instead of elementary algebra where the values of the variables are numbers, and the main operations are addition and multiplication,  The main operations of Boolean algebra are the conjunction and denoted as ∧, the disjunction or denoted as ∨, and the negation not deoted as¬. It is thus a formalism for describing logical relations in the same way that ordinary algebra describes numeric relations 4-4  Boolean algebra satisfies many of the same laws as ordinary algebra when one matches up ∨ with addition and ∧ with multiplication. In particular the following laws are common to both kinds of algebra 4-5 Boolean functions and truth tables The basic functions of Boolean calculus are as follows. AND (conjunction), denoted x∧y (sometimes x AND y or Kxy), satisfies x∧y = 1 if x = y = 1 and x∧y = 0 otherwise. OR (disjunction), denoted x∨y (sometimes x OR y or Axy), satisfies x∨y = 0 if x = y = 0 and x∨y = 1 otherwise. NOT (negation), denoted ¬x (sometimes NOT x, Nx or !x), satisfies ¬x = 0 if x = 1 and ¬x = 1 if x = 0. 4-6 Truth tables The values of x∧y, x∨y, and ¬x can be expressed by tabulating their values with truth tables 4-7 4-8 Secondary Functions 4-9 Combinational logic 4-10 Combinational logic 4-11 Example of ARC Memory Layout 4-12 4-13 Try Yourself Draw a truth table for the following expressions: 1.A +B 2.AB 3.A + BC 4.A (B + D) 5.(A + B)(A + C) 6.(A + B)(A + C) 4-14 Try Yourself Draw a logic circuit for the following expressions: 1.(A + B)C 2.A + BC + D 3.AB + AC 4.(A + B)(C + D)C 4-15 SUMMARY o The fundamental concept of Boolean algebra, Boolean functions, truth tables, algebra manipulation, Boolean expressions and simplification of Boolean Functions o The correspondence between electronic circuits and boolean functions, combinational logic and sequential Logic o Steps to draw the truth table and circuit diagram. 4-16 THE END