Chapter 3 Boolean Algebra and Digital Logic Chapter 3 Objectives Understand the relationship between Boolean logic and digital computer circuits Learn how to design simple logic circuits Understand how digital circuits work together to form complex computer systems 2 3 1 Introduction In the latter part of the nineteenth century George Boole incensed philosophers and mathematicians alike when he suggested that logical thought could be represented through mathematical equations How dare anyone suggest that human thought could be encapsulated and manipulated like an algebraic formula Computers as we know them today are implementations of Boole s Laws of Thought John Atanasoff and Claude Shannon were among the first to see this connection 3 3 1 Introduction In the middle of the twentieth century computers were commonly known as thinking machines and electronic brains Many people were fearful of them Nowadays we rarely ponder the relationship between electronic digital computers and human logic Computers are accepted as part of our lives Many people however are still fearful of them In this chapter you will learn the simplicity that constitutes the essence of the machine 4 3 2 Boolean Algebra Boolean algebra is a mathematical system for the manipulation of variables that can have one of two values In formal logic these values are true and false In digital systems these values are on and off 1 and 0 or high and low Boolean expressions are created by performing operations on Boolean variables Common Boolean operators include AND OR and NOT 5 3 2 Boolean Algebra A Boolean operator can be completely described using a truth table The truth table for the Boolean operators AND and OR are shown at the right The AND operator is also known as a Boolean product The OR operator is the Boolean sum 6 3 2 Boolean Algebra The truth table for the Boolean NOT operator is shown at the right The NOT operation is most often designated by an overbar It is sometimes indicated by a prime mark or an elbow 7 3 2 Boolean Algebra A Boolean function has At least one Boolean variable At least one Boolean operator and At least one input from the set 0 1 It produces an output that is also a member of the set 0 1 Now you know why the binary numbering system is so handy in digital systems 8 3 2 Boolean Algebra The truth table for the Boolean function is shown at the right To make evaluation of the Boolean function easier the truth table contains extra shaded columns to hold evaluations of subparts of the function 9 3 2 Boolean Algebra As with common arithmetic Boolean operations have rules of precedence The NOT operator has highest priority followed by AND and then OR This is how we chose the shaded function subparts in our table 10 3 2 Boolean Algebra Digital computers contain circuits that implement Boolean functions The simpler that we can make a Boolean function the smaller the circuit that will result Simpler circuits are cheaper to build consume less power and run faster than complex circuits With this in mind we always want to reduce our Boolean functions to their simplest form There are a number of Boolean identities that help us to do this 11 3 2 Boolean Algebra Most Boolean identities have an AND product form as well as an OR sum form We give our identities using both forms Our first group is rather intuitive 12 3 2 Boolean Algebra Our second group of Boolean identities should be familiar to you from your study of algebra 13 3 2 Boolean Algebra Our last group of Boolean identities are perhaps the most useful If you have studied set theory or formal logic these laws are also familiar to you 14 3 2 Boolean Algebra We can use Boolean identities to simplify the function as follows 15 3 2 Boolean Algebra Sometimes it is more economical to build a circuit using the complement of a function and complementing its result than it is to implement the function directly DeMorgan s law provides an easy way of finding the complement of a Boolean function Recall DeMorgan s law states 16 3 2 Boolean Algebra DeMorgan s law can be extended to any number of variables Replace each variable by its complement and change all ANDs to ORs and all ORs to ANDs Thus we find the the complement of is 17 3 2 Boolean Algebra Through our exercises in simplifying Boolean expressions we see that there are numerous ways of stating the same Boolean expression These synonymous forms are logically equivalent Logically equivalent expressions have identical truth tables In order to eliminate as much confusion as possible designers express Boolean functions in standardized or canonical form 18 3 2 Boolean Algebra There are two canonical forms for Boolean expressions sum of products and product of sums Recall the Boolean product is the AND operation and the Boolean sum is the OR operation In the sum of products form ANDed variables are ORed together For example In the product of sums form ORed variables are ANDed together For example 19 3 2 Boolean Algebra It is easy to convert a function to sum of products form using its truth table We are interested in the values of the variables that make the function true 1 Using the truth table we list the values of the variables that result in a true function value Each group of variables is then ORed together 20 3 2 Boolean Algebra The sum of products form for our function is We note that this function is not in simplest terms Our aim is only to rewrite our function in canonical sum of products form 21 3 3 Logic Gates We have looked at Boolean functions in abstract terms In this section we see that Boolean functions are implemented in digital computer circuits called gates A gate is an electronic device that produces a result based on two or more input values In reality gates consist of one to six transistors but digital designers think of them as a single unit Integrated circuits contain collections of gates suited to a particular purpose 22 3 3 Logic Gates The three simplest gates are the AND OR and NOT gates They correspond directly to their respective Boolean operations as you can see by their truth tables 23 3 3 Logic Gates Another very useful gate is the exclusive OR XOR gate The output of the XOR operation is true only when the values of the inputs differ Note the special symbol for the XOR operation 24 3 3 Logic Gates NAND and NOR are two very important gates Their symbols and truth tables are shown at the right 25 3 3 Logic Gates NAND and NOR are known as universal gates because they are inexpensive