Lecture 7 Topics –Boolean Algebra 1

Logic and Bits Operation Computers represent information by bit A bit has two possible values, namely zero and one. A bit can be used to represent a truth value, since there are two truth values, true and false. Bits operations correspond to the logical operations in Boolean Algebra. 2

Boolean Variables & Boolean Operators Boolean variables are variables that can take only binary values: 0 or 1, false or true –A,B,C = {0, 1} Boolean Operators –AND (A AND B, AB, A ∧ B) –OR (A OR B, A+B, A ∨ B) –NOT (NOT A, A') 3

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 Boolean operations on Boolean variables. –Common Boolean operators include AND, OR, and NOT. 4

5 A Boolean operator can be completely described using a truth table. The AND operator is also known as a Boolean product. The OR operator is the Boolean sum. Truth Table of Boolean Operators

Boolean Function 6 To make evaluation of the Boolean function easier, the truth table contains extra (shaded) columns to hold evaluations of subparts of the function.

7 Order of Boolean Operations There might be many Boolean operators in one Boolean function. Which operator to apply first? Order of Boolean Operations: NOT > AND > OR

In Class Exercise 8

9 Boolean Identities: Simplify a Boolean Function 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 simplify a Boolean function.

10 Boolean Identities

How to prove these identities All of the above identities can be proved using truth tables. To do this, you use truth tables to show all of the possible values of both sides of the equation. If they are identical, then the identity is true. 11

Proving the AND Form of DeMorgan’s Law 12

Simplification of Boolean Expressions 13

Simplification of Boolean Expressions 14

Simplification of Boolean Expressions 15

16 Example 3.6 Simplify the function: Simplification of Boolean Expressions

Boolean Algebra Properties

In Class Exercise 18

19 Complements Sometimes it is more economical to build a circuit using the complement of a function 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:

20 Complements

In Class Exercise 21

Representing Boolean Functions 22

23 Sum-of-Products Form

In Class Exercise The true table for a Boolean expression is shown below. Write the Boolean expression in sum-of-products form. 24

Summary Given a boolean function, construct it’s truth table. Using truth table to prove a boolean equation is valid or not. Applying boolean algebra for boolean function simplification. Using DeMorgan’s Law, write an expression for the complement. Determine the boolean expression in sum-of- products form. 25