Boolean Algebra and Logic Gates

F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, self assessment programme for students who wish to understand the basic techniques of logic gates.

c 2005 Email: chamer, mlavelle, dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0

Table of Contents

1. 2. 3. 4. 5. Logic Gates (Introduction) Truth Tables Basic Rules of Boolean Algebra Boolean Algebra Final Quiz Solutions to Exercises Solutions to Quizzes

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Section 1: Logic Gates (Introduction)

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1. Logic Gates (Introduction)

The package Truth Tables and Boolean Algebra set out the basic principles of logic. Any Boolean algebra operation can be associated with an electronic circuit in which the inputs and outputs represent the statements of Boolean algebra. Although these circuits may be complex, they may all be constructed from three basic devices. These are the AND gate, the OR gate and the NOT gate. x y AND gate x·y x y OR gate x+y x NOT gate x

In the case of logic gates, a diﬀerent notation is used: x ∧ y, the logical AND operation, is replaced by x · y, or xy. x ∨ y, the logical OR operation, is replaced by x + y. ¬x, the logical NEGATION operation, is replaced by x or x. The truth value TRUE is written as 1 (and corresponds to a high voltage), and FALSE is written as 0 (low voltage).

Section 2: Truth Tables

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2. Truth Tables

x y x·y

x 0 0 1 1 Summary

y x·y 0 0 1 0 0 0 1 1 of AND gate

x 0 0 1 1 Summary

y x+y 0 0 1 1 0 1 1 1 of OR gate

x y

x+y

x

x

x 0 1 Summary of

x 1 0 NOT gate

Section 3: Basic Rules of Boolean Algebra

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3. Basic Rules of Boolean Algebra

The basic rules for simplifying and combining logic gates are called Boolean algebra in honour of George Boole (1815 – 1864) who was a self-educated English mathematician who developed many of the key ideas. The following set of exercises will allow you to rediscover the basic rules: x Example 1 1 Consider the AND gate where one of the inputs is 1. By using the truth table, investigate the possible outputs and hence simplify the expression x · 1. Solution From the truth table for AND, we see that if x is 1 then 1 · 1 = 1, while if x is 0 then 0 · 1 = 0. This can be summarised in the rule that x · 1 = x, i.e., x x 1

Section 3: Basic Rules of Boolean Algebra

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Example 2 x 0 Consider the AND gate where one of the inputs is 0. By using the truth table, investigate the possible outputs and hence simplify the expression x · 0. Solution From the truth table for AND, we see that if x is 1 then 1 · 0 = 0, while if x is 0 then 0 · 0 = 0. This can be summarised in the rule that x · 0 = 0 x 0 0

Section 3: Basic Rules of Boolean Algebra

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Exercise 1. (Click on the green letters for the solutions.) Obtain the rules for simplifying the logical expressions x (a) x + 0 which corresponds to the logic gate 0 (b) x + 1 which corresponds to the logic gate x 1

Exercise 2. (Click on the green letters for the solutions.) Obtain the rules for simplifying the logical expressions: x (a) x + x which corresponds to the logic gate

(b) x · x which corresponds to the logic gate

x

Section 3: Basic Rules of Boolean Algebra

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Exercise 3. (Click on the green letters for the solutions.) Obtain the rules for simplifying the logical expressions: (a) x + x which corresponds to the logic gate x

(b) x · x which corresponds to the logic gate

x

Quiz Simplify the logical expression (x ) represented by the following circuit diagram. x (a) x (b) x (c) 1 (d) 0

Section 3: Basic Rules of Boolean Algebra

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Exercise 4. (Click on the green letters for the solutions.) Investigate the relationship between the following circuits. Summarise your conclusions using Boolean expressions for the...