Page 1 Basic Engineering Boolean Algebra and Logic Gates F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, selfassessmentprogrammeforstudentswhowish tounderstandthebasictechniquesoflogicgates. c 2005 Email: chamer,mlavelle,dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0 Page 2 Basic Engineering Boolean Algebra and Logic Gates F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, selfassessmentprogrammeforstudentswhowish tounderstandthebasictechniquesoflogicgates. c 2005 Email: chamer,mlavelle,dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0 Table of Contents 1. Logic Gates (Introduction) 2. Truth Tables 3. Basic Rules of Boolean Algebra 4. Boolean Algebra 5. Final Quiz Solutions to Exercises Solutions to Quizzes Thefullrangeofthesepackagesandsomeinstructions, shouldtheyberequired,canbeobtainedfromourweb page Mathematics Support Materials. Page 3 Basic Engineering Boolean Algebra and Logic Gates F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, selfassessmentprogrammeforstudentswhowish tounderstandthebasictechniquesoflogicgates. c 2005 Email: chamer,mlavelle,dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0 Table of Contents 1. Logic Gates (Introduction) 2. Truth Tables 3. Basic Rules of Boolean Algebra 4. Boolean Algebra 5. Final Quiz Solutions to Exercises Solutions to Quizzes Thefullrangeofthesepackagesandsomeinstructions, shouldtheyberequired,canbeobtainedfromourweb page Mathematics Support Materials. Section 1: Logic Gates (Introduction) 3 1. Logic Gates (Introduction) ThepackageTruthTablesandBooleanAlgebrasetoutthebasic 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 x·y AND gate x y x+y OR gate x x 0 NOT gate In the case of logic gates, a di?erent 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 0 or x. The truth value TRUE is written as 1 (and corresponds to a high voltage), and FALSE is written as 0 (low voltage). Page 4 Basic Engineering Boolean Algebra and Logic Gates F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, selfassessmentprogrammeforstudentswhowish tounderstandthebasictechniquesoflogicgates. c 2005 Email: chamer,mlavelle,dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0 Table of Contents 1. Logic Gates (Introduction) 2. Truth Tables 3. Basic Rules of Boolean Algebra 4. Boolean Algebra 5. Final Quiz Solutions to Exercises Solutions to Quizzes Thefullrangeofthesepackagesandsomeinstructions, shouldtheyberequired,canbeobtainedfromourweb page Mathematics Support Materials. Section 1: Logic Gates (Introduction) 3 1. Logic Gates (Introduction) ThepackageTruthTablesandBooleanAlgebrasetoutthebasic 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 x·y AND gate x y x+y OR gate x x 0 NOT gate In the case of logic gates, a di?erent 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 0 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 4 2. Truth Tables x y x·y x y x·y 0 0 0 0 1 0 1 0 0 1 1 1 Summary of AND gate x y x+y 0 0 0 0 1 1 1 0 1 1 1 1 Summary of OR gate x y x+y x x 0 x x 0 0 1 1 0 Summary of NOT gate Page 5 Basic Engineering Boolean Algebra and Logic Gates F Hamer, M Lavelle & D McMullan The aim of this document is to provide a short, selfassessmentprogrammeforstudentswhowish tounderstandthebasictechniquesoflogicgates. c 2005 Email: chamer,mlavelle,dmcmullan@plymouth.ac.uk Last Revision Date: August 31, 2006 Version 1.0 Table of Contents 1. Logic Gates (Introduction) 2. Truth Tables 3. Basic Rules of Boolean Algebra 4. Boolean Algebra 5. Final Quiz Solutions to Exercises Solutions to Quizzes Thefullrangeofthesepackagesandsomeinstructions, shouldtheyberequired,canbeobtainedfromourweb page Mathematics Support Materials. Section 1: Logic Gates (Introduction) 3 1. Logic Gates (Introduction) ThepackageTruthTablesandBooleanAlgebrasetoutthebasic 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 x·y AND gate x y x+y OR gate x x 0 NOT gate In the case of logic gates, a di?erent 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 0 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 4 2. Truth Tables x y x·y x y x·y 0 0 0 0 1 0 1 0 0 1 1 1 Summary of AND gate x y x+y 0 0 0 0 1 1 1 0 1 1 1 1 Summary of OR gate x y x+y x x 0 x x 0 0 1 1 0 Summary of NOT gate Section 3: Basic Rules of Boolean Algebra 5 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: Example 1 x 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 1 xRead More

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