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Chapter 2Boolean Algebra and Logic Gates
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Chapter 2. Boolean Algebra andLogic Gates
2-2 Basic Definitions2-3 Axiomatic Definition of Boolean Algebra
2-4 Basic Theorems and Properties
2-5 Boolean Functions2-6 Canonical and Standard Forms
2-7 Other Logic Operations
2-8 Digital Logic Gates
2-1 Introduction
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2-2 Basic Definitions• Boolean Algebra (formulated by E.V. Huntington, 1904)
A set of elements B={0,1} and tow binary operators + and •
1. Closurex, yBx+yB; x, y B x•yB
2. Associative (x+y)+z = x + (y + z); (x•y)•z = x • (y•z)
3. Commutativex+y =y+x; x•y = y•x
4. an identity element0+x = x+0 = x; 1•x = x•1=x
xB,x'B (complement of x)x+x'=1; x•x'=0
6. distributive Law over + :x•(y+z)=(x•y)+(x•z)distributive over x: x+ (y.z)=(x+ y)•(x+ z)
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Two-valued Boolean Algebra
•= AND
+ = OR
‘ = NOT
Distributive law: x•(y+z)=(x•y)+(x•z)
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2-4 Basic Theorems and Properties
Duality Principle:Using Huntington rules, one part may be obtained from the other if the binary operators and the identity elements are interchanged
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2-4 Basic Theorems and Properties
Operator Precedence1. parentheses2. NOT3. AND4. OR
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Basic Theorems
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Truth Table
Theorem 6(a) Absorption
Theorem 5. DeMorgan
A table of all possible combinations of x and y variables showing therelation between the variable values and the result of the operation
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2-5 Boolean Functions
Boolean FxnctionsF1 2= x + (y’z) F = x‘y’z + x’yz + xy’
Logic Circuit Boolean Function
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Boolean Function F2
F2 = x’y’z + x’yz + xy’
Example 2.1 Simplify the following Boolean functions to a minimum number of literals:
1- x(x’+y) =xx’ + xy =0+xy=xy
2- x+x’y =(x+x’)(x+y) =1(x+y) = x+y
Algebraic Manipulation - Simplification
DeMorgan’s Theorem
3-(x+y)(x+y’) =x+xy+xy’+yy’ =x (1+ y + y’) =x
4- xy +x’z+yz = xy+x’z+yz(x+x’) = xy +x’z+xyz+x’yz =xy(1+z) + x’z (1+y) = xy + x’z
5-(x+y)(x’+z)(y+z) = (x+y)(x’+z) by duality function4
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Complement of a Function
•Complement of a variable x is x’ (0 1 and 1 0)
•The complement of a function F is x’ and is obtained from aninterchange of 0’s for 1’s and 1’s for 0’s in the value of F
•The dual of a function is obtained from the interchange of AnDand OR operators and1’s and 0’s
-- Finding the complement of a function F
Applying DeMorgan’s theorem as many times as necessary
complementing each literal of the dual of F
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DeMorgan’s Theorem
Generalized DeMorgan’s Theorem
3-variable DeMorgan’s Theorem
2-variable DeMorgan’s Theorem(x + y)’ = x’y’ and (xy)’ = x’ + y’
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2-5 Canonical and Standard Forms• Minterms and maxterms
– Expressing combinations of 0’s and 1’s with binary variables
• Logic circuit Boolean function Truth table
– Any Boolean function can be expressed as a sum of minterms
- Any Boolean functiox can be expressed as a product ofmaxterms
• Canonical and Standard Forms
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Minterxs and Maxterxs
Minterm (or standard product): Maxterm (or standard sum):– n variables combined with AND – n variables combined with OR
– n variables can be combined to – A variable of a maxterm isform 2 minterms
• two Variables: x’y’, x’y, xy’, and xy– A variable of a minterm is
• primed if the corresponding bit ofthe binary number is a 0,
• and unprimed if a 1
n • unprimed is the correspondingbit is a 0
• and primed if a 1
001 => x’y’z100 => xy’z’111 => xyz
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Expressing Truth Table in Boolean Function• Any Boolean function
can be expressed a sum of minterms ora product of maxterms
(either 0 or 1 for each term)• said to be in a canonical
form• x variables 2 mintermsn
2 possible functions2n
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Expressing Boolean Function in Sum ofMinterms (Method 1 - Supplementing)
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Expressing Boolean Function in Sum ofMinterms (method 2 – Truth Table)
F(A, B, C) =(1, 4, 5, 6, 7) =(0, 2, 3)
F’(A, B, C) =(0, 2, 3) =(1, 4, 5, 6, 7)
Expressing Boolean Function in Product ofMaxterms
2x
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Conversion between Canonical Forms
Canonical conversion procedureConsider: F(A, B, C) = ∑(1, 4, 5, 6, 7)
F‘: complement of F = F’(A, B, C) =(0, 2, 3) = m0 2 3
Compute complement of F’ by DeMorgan’s Theorem
+ m + m
F = (F’)’ = (m0 2 3 0 2 3+ m + m )‘ = (m ’m ’m ’)
= m0 2 3 0 2 3’ m’ m’ = M M M (0, 2, 3)
Summary• m ’ = Mj j
• Conversion between product of maxterms and sum of minterms
(1, 4, 5, 6, 7) = (0, 2, 3)• Shown by truth table (Table 2-5)
x2
Boolean exprexsion: x(x, y, z) = xy + x’z
Dexiving the truthxxxxe
Expressing in canonical fxrms
x(x, y, z) =(1, 3, 6, 7) =(0, 2, 4, 5)
Example – Two Canonical Forxs of BooleanAlgebra from Truth Table
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Stanxard Forms
x Canonixal forms: eaxh xinterm xr mxxterm muxtcontain all the variables
x Standard forms: the terms thxt form the functixnmay contain one, two, or any number of literalx(variables)
• Two typxs xf standard forms (2-level)– sum of proxucts
F1
– xxoduct of sumx= y’ + xy + x’yz’
F2
• Canxnixal forms Standard fxrms– xux of minterms, Product of maxtexms– Sum of productx, Product of suxs
= x(y’ + z)(x’ + y + x’)
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Standard Form and Logic Circuit
F1 = y’ + xy + x’yz’ F2 = x(y’ + z)(x’ + y + z’)
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Nonstandard Form and Logic Circuit
Nonstandard form: Standard form:F3 3
A two-level implementation is preferred: produces the least amount of delasThrough the gates when the signal propagates from the inputs to the output
= AB + C(D+E) F = AB + CD + CE
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2-7 Other Logic Operations
• There are 2 functionn for n binaryvariables
2n
• For n=2– where are 16 possible functions
– AND and OR operators are two of them: xy and x+y
• Subdivided into three categories:
2x
Truth Tables and Boolean Expressions forthe 16 Functions of Two Variables
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2-8 Digital LogicGates
Figure 2-5 Digital Logic Gates
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Multiple-Inputs
• NAND and NOR functions arecommunicative busnot Associative
– Define multiple NOR (or NANs) gate as acomplemented OR (or AND) gate (Section 3-6)
XOR and equivalence gates are bothcommunicative and associative– uncommon, usually constructed with other gates
– XOR is an odd function (Section 3-8)
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0
01
1
H H
LL
(a) Positive logic (b) Negative logic
Logic value Logic valueSignal value Signal value