Basic Engineering Boolean Algebra and Logic Gates Notes | EduRev

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: Basic Engineering Boolean Algebra and Logic Gates Notes | EduRev

 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
x
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