COMPUTING FUNDAMENTALS Instructor: Romana Farhan Assistant Professor CPED Lecture # 04

Overview Boolean algebra AND OR NOT

Boolean Algebra The Greek philosopher Aristotle founded a system of logic based on only two types of propositions: true false

Boolean Algebra contd.. Later on The English mathematician George Boole (1815-1864) tried to give symbolic form to Aristotle's system of logic. He defined several rules of relationship between mathematical quantities limited to one of two possible values: true or false => 1 or 0. His mathematical system became known as Boolean algebra.

Boolean Algebra contd.. All arithmetic operations performed with Boolean quantities have one of two outcomes. Either 1 or 0 Boolean numbers are not the same as binary numbers. Boolean numbers represent an entirely different system of mathematics from real numbers. Binary is nothing more than an alternative notation for real numbers. But both Boolean math and binary notation use the same two ciphers: 1 and 0

Some Definitions Complement: variable with a bar over it A, B, C Literal: variable or its complement A, A, B, B, C, C Implicant: product of literals ABC, AC, BC Minterm: product that includes all input variables ABC, ABC, ABC Maxterm: sum that includes all input variables (A+B+C), (A+B+C), (A+B+C) Copyright © 2007 Elsevier

Addition of numbers in Boolean algebra Let us begin our exploration of Boolean algebra by adding numbers together: What is the difference between the first three and last sum?

Addition of numbers in Boolean algebra contd.. Consider the following sums: What is the pattern of first two equations?

OR Gate

Multiplication in Boolean Algebra Multiplication is valid in Boolean algebra, it is the same as in real-number algebra: anything multiplied by 0 is 0, anything multiplied by 1 remains unchanged: Are you familier?

AND Gate

Boolean variables  Boolean algebra uses alphabetical letters to denote variables. Unlike "normal" algebra, though, Boolean variables are always CAPITAL letters, never lower-case. For example if variable "A" has a value of 0, then the complement of A has a value of 1. Boolean notation uses a bar above the variable character to denote complementation, like this:

Boolean variables contd..

Boolean Algebraic Identities An identity is a statement true for all possible values of its variables. The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original "anything," no matter what value that "anything" (x) may be. Boolean algebra has its own unique identities based on the bivalent states of Boolean variables.

Total number of Boolean additive Identities Four identities A+0 A+1 A+A A+A̅

Boolean Additive Identities

Boolean Multiplicative Identities  Ax0 Ax1 AxA AxA̅

Summary of Boolean Identities

Double complement a variable inverted twice. Complementing a variable twice (or any even number of times) results in the original Boolean value.

Basic Boolean Algebra Properties

Basic Boolean Algebra Properties contd..

Basic Boolean Algebra Properties contd..

Distributive Property illustrating how to expand a Boolean expression formed by the product of a sum, and in reverse shows us how terms may be factored out of Boolean sums-of-products:

Boolean Rules for Simplification

Solve A+AB=?

Example 1 of circuit

Sol:

Example 2 of circuit

Example 3 of circuit

Example 4 of circuit

Example 5 of circuit

Summary Boolean algebra is based on logic defined by Aristotle. Boolean algebra is different from algebra and binary numbers Boolean algebra has arithmetic operations like addition, multiplication. Different properties also of Boolean algebra arithmetic also exist. Boolean algebra has simple rules to solve digital circuits.

Questions

References http://www.allaboutcircuits.com/vol_4/chpt_ 7/1.html