ch2 boolean algebra
TRANSCRIPT
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Boolean Algebra &
Logic Gates
ECE 223Digital Circuits and Systems
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Binary (Boolean) Logic
Deals with binary variables and binary logicfunctions
Has two discrete values
0 False, Open
1 True, Close
Three basic logical operations
AND (.); OR (+); NOT ()
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Logic Gates & Truth Tables
AND ORNOT
101001
1
0
0
A
OR
1
1
0
B
1
1
0
A+B
1
0
A
1
0
0
A
AND
1
1
0
B
1
0
0
A.B
0
1
A
NOT
AND; OR gates may have any # of inputs
AND 1 if all inputs are 1; 0 other wise
OR 1 if any input is 1; 0 other wise
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Boolean Algebra
Branch of Algebra used for describing and designingtwo valued state variables Introduced by George Boole in 19th centaury Shannon used it to design switching circuits (1938)
Boolean Algebra Postulates
An algebraic structure defined by a set of elements, B,together with two binary operators + and . that satisfy thefollowing postulates:
1. Postulate 1:Closure with respect to both (.) and ( +)
2. Postulate 2:An identity element with respect to +, designated by 0. An identity
element with respect to . designated by 1
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Boolean Algebra - Postulates
3. Postulate 3:Commutative with respect to + and .
4. Postulate 4:
Distributive over . and +
5. Postulate 5:
For each element a of B, there exist an element a such that(a) a + a = 1 and (b) a.a = 0
6. Postulate 6:
There exists at least two elements a, b in B, such that a b
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Boolean Algebra - Postulates
Postulates are facts that can be taken as true; they donot require proof
We can show logic gates satisfy all the postulates
101001
1
0
0
A
OR
1
1
0
B
1
1
0
A+B
1
0
A
1
0
0
A
AND
1
1
0
B
1
0
0
A.B
0
1
A
NOT
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Boolean Algebra - Theorems
Theorems help us out in manipulating Booleanexpressions They must be proven from the postulates and/or other already
proven theorems
Exercise Prove theorems from postulates/other proventheorems
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Boolean Functions
Are represented as
Algebraic expressions;
F1 = x + yz Truth Table
Synthesis
Realization of schematic from theexpression/truth table
Analysis Vice-versa
10011101
0110
1011
1
0
0
0
x
001
1
0
0
y
1
1
0
z
1
1
0
F1
x
y
z
F1
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Synthesis F1
Assume true as well as complement inputsare available
Cost
A 2-input AND gate
A 2-input OR gate
4 inputs1001
1101
0110
1011
1
0
0
0x
001
1
0
0y
1
1
0z
1
1
0F1
x
y
z
F1
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Canonical and Standard Forms
Minterms A minterm is an AND term in which every literal
(variable) of its complement in a function occurs once
For n variable 2n minterms
Each minterm has a value of 1 for exactly one
combination of values of n variables(e.g., n = 3)
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Minterms
One method of Writing Boolean function is thecanonical minterm (sum of products or SOP) form F = xyz +xyz + xyz = m1 + m5 + m6 = (1,5,6)
xyz
xyz
xyz
xyz
xyz
xyz
xyz
xyz
Corresponding
minterm
m4001
m5101
m3110
m6011
1
0
0
0
x
m201
1
0
0
y
1
1
0
z
m7
m1
m0
Designation
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Minterms examples
F2 = (0,1,2,3,5)
= xyz + xyz + xyz + xyz + xyz
0
0
1
0
1
1
1
1
F2 (Given)
001
m5101
m3110
011
1
0
0
0
x
m201
1
0
0
y
1
1
0
z
m1
m0
Designation
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Minterms examples
(F2) = (all minterms not in F2) = (4,6,7)
= xyz + xyz + xyz
0
0
1
0
1
1
1
1
F2 (Given)
001
m5101
m3110
011
1
0
0
0
x
m201
1
0
0
y
1
1
0
z
m1
m0
Designation
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Maxterms
A maxterm is an OR term in which every literal(variable) or its complement in a function occurs once Each maxterm has a value 0 for one combination of values of
n variables
x +y +z
x +y +z
x +y +z
x +y +z
x +y +z
x +y +z
x +y +z
x +y +z
Corresponding
maxterm
M4001
M5101
M3110
M6011
1
0
0
0
x
M201
1
0
0
y
1
1
0
z
M7
M1
M0
Designation
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Minterms & Maxterms
Conversion between minterms & maxtermsm0 = xyz = (x+y+z) = (M0)
In general, mi = (Mi)
An alternative method of writing a Boolean function isthe canonical maxterm (product of sums or POS) form
The canonical product of sums can be written directlyfrom the truth table
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Maxterms
F3 = (x+y+z)(x+y+z)(x+y+z)(x+y+z)(x+y+z)= (0,2,3,4,7)
(F3) = (all maxterm not in F3)
0
1
1
0
0
0
1
0
F3 (Given)
M4001
101
M3110
011
1
0
0
0
x
M201
1
0
0
y
1
1
0
z
M7
M0
Designation
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Standard Forms
In canonical forms, each minterm (or maxterm)must contain all variables (or its complements)The algebraic expressions can further be simplified
ExampleF4 (x,y,z) = xy +yz (sum of products, standard form)
F5 (x,y,z) = (x+y)(y+z) (product of sums, standard form) Conversion
Standard form can be converted into canonical form using identity elements
F4 = xy + yz = xy.1 +1.yz = xy(z+z) + (x+x)yz
= xyz + xyz + xyz + xyz = m7 +m6 +m5 +m1
How about the conversion from canonical forms tostandard forms?
Exercise convert F5 into maxterms
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Non-Standard Forms
A Boolean function may be written in non-standard formF6 (x,y,z) = (xy + z)(xz + yz)
= xy(xz + yz) + z(xz + yz)
= xyz + xyyz + xz +yz
= xyz + xz + yz
= xz + yz (standard form)
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Other Logic Gates NAND Gate So far, we discussed AND, OR, NOT gates
2-input NAND (NOT-AND operation)
Can have any # of inputs
NAND gate is not associative
Associative property to be discussed later
x
y
z
011
1
0
0
x
10
1
0
y
1
1
z
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Other Logic Gates NOR Gate 2-input NOR (NOT-OR operation)
Can have any # of inputs
NOR gate is not associative
Associative property to be discussed later
x
y
z
011
1
0
0
x
00
1
0
y
0
1
z
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Other Logic Gates XOR Gate 2-input XOR
Output is 1 if any input is one and the other input is 0
Can have any # of inputs
x
y
z
011
1
0
0
x
10
1
0
y
1
0
z
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Other Logic Gates XNOR Gate 2-input XNOR
Performs the NOT-XOR operation
Output is 1 if both inputs are 1; or both inputs are 0
Can have any # of inputs
x
y
z
111
1
0
0
x
00
1
0
y
0
1
z
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Extension to Multiple Inputs So far, we restricted ourselves to 1 or 2-input gates
A logic gate (except inverter) can have any number of inputs
AND, OR logic operations have two properties
x +y = y +x (commutative)
(x +y)+ z = x + (y +z) = x +y +z (associative)
NAND and NOR operations are commutative, but notassociative
(xy)z x(yz) = NOR operation
(xy)z x(yz) = NAND operation
How about XOR
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Positive & Negative Logic
Positive Logic 0 = False (Low Voltage)
1 = True (High Voltage)
Negative Logic
0 = True (High Voltage)
1 = False (Low Voltage) Implement truth table with
positive & negative logic
Positive logic AND gate
Negative logic ?
HHH
H
L
L
x
LL
H
L
y
L
L
z
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Integrated Circuit - Evolution
Transistor was invented in 1947/48
Integrated Circuits were invented in 1959/60
Since then, larger # of transistors/chip are integrated
101 Small Scale Integration
102-3 Medium Scale Integration
103-6 Large Scale Integration
106-9 Very Large Scale Integration
Digital Logic Families (technologies)
TTL Transistor-Transistor Logic
ECL Emitter Coupled Logic
MOS Metal Oxide Semiconductor
CMOS Complementary Metal Oxide Semiconductor
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Book Sections Boolean Algebra & Logic
Gates Material is covered in Sections 2.1 2.8