Chapter 2 (Lect 2)
•Canonical and Standard Forms
•Sum of Minterms
•Product of Maxterms
•Standard Form
•Sum of products
•Product of sums
•Other Logic Operators
•Logic Gates
•Basic and Multiple Inputs
•Positive and Negative Logic
•Integrated Circuits
Sum of minterms: from truth table
1. Indentify minterms that equate to 1
2. OR them together to form Boolean expression
x y
z
F
0 0
0
0
0 0
1
1
0 1
0
0
0 1
1
1
1 0
0
0
1 0
1
0
1 1
0
1
1 1
1
0
F=
Minterm
Designation
F(x.y,x) = ∑
Sum of minterms: from function
In canonical form each minterm contains all variables, primed or unprimed
1. Expand into a sum of AND terms
2. Treat each term separately and add in missing terms by ANDing with
(x + x’) = 1
3. Recombine remove duplicates
F = A + B’C
Or create a truth table
Product of maxterms: from truth table
1. Identify maxterms that evaluate to 0
2. AND them together to form Boolean expression
x
y
z
F
0
0
0
0
0
0
1
1
0
1
0
0
0
1
1
1
1
0
0
0
1
0
1
0
1
1
0
1
1
1
1
0
F=
Maxterm
Designation
F(x.y,x) =∏
Product of maxterms: from function
In canonical form each maxterm contains all variables, primed or unprimed
1. Expand into ANDed -ORed terms using x + yz = (x + y)(x + z) repeatedly
2. Treat each term separately, and add in missing terms by ORing (xx’) = 0
3. Recombine remove duplicates
F = AB + B’C
Or create a truth table
Conversion Between Canonical Forms
1. Interchange the symbols ∑ and ∏
2. List missing numbers in parenthesizes
3. This is the other canonical form
F (x, y, z)= ∑(m0, m2,m4,m7)
F = x’ + yz
Does SOP and POS produce the same result?
x y
F
0 0
0
0 1
1
1 0
1
1 1
0
Minterm
Maxterm
Canonical form: Sum of minterms
F = xyz + x’yz + xyz’
Standard form: Sum-of-products SOP
F = yz + xy
Canonical form: Product of maxterms
F = (x + y + z)(x’ + y + z’)
Standard form: Products-of-sums POS
F = x(x + y’)(y’ + z)
Non-Standard form
F = x+x(y’ + z)
Creating circuit from expression
F = x’y + xy’ Make note AND-OR implementation
F = (x + y’) (x’ + y) Make note OR-AND implementation
Most used Boolean functions and Operators
(From Table 2.8)
Function
Operator Symbol
F=x
Name/Comments
Transfer, buffer
F = x’
x’
Complement, not x
F = xy
x·y
AND, x and y
F = (xy)’
x↑y
NAND, not AND
F = x+y
x+y
OR,
F = (x+y)’
x↓y
NOR,
x or y
not OR
F = xy’ + x’y
x
⊕
y
Exclusive OR,
F = xy + x’y’
(x
⊕
y)’
XNOR,
x=y
x or y not both
Logic Gates
Buffer
Inverter
F=x
F = x’
x
F
0
0
1
1
x
F
0
1
1
0
Logic Gates
AND
OR
F = xy
F=x+y
x
y
F
0
0
0
0
1
0
1
0
0
1 1
1
x
y
F
0
0
0
0
1
1
1
0
1
1 1
1
NAND
NOR
x
y
F
F = (xy)’
0
0
1
= x↑y
0
1
1
1
0
1
1 1
0
x
y
F
F = (x + y)’
0
0
1
= x↓y
0
1
0
1
0
0
1 1
0
XOR
F = xy’ + x’y
=
XNOR
x⊕y
F = xy + x’y’
=
(x ⊕ y)'
x
y
F
0
0
0
0
1
1
1
0
1
1 1
0
x
y
F
0
0
1
0
1
0
1
0
0
1 1
1
Extension to Multiple inputs (more than two)
Must satisfy Commutative and Associative laws
AND and OR gates no problem
NOR and NANDs ok if we use
(OR)’ ….. (x + y + z)’
(AND)’ …. (xyz)’
Example1
Works for XOR, with modified definition, But uncommon in hardware
implementation
x
y
T1
F1
z
x
T2
y
z
F2
x
y
z
0
0
0
0
0
1
0
1
0
0
1
1
1
0
0
1
0
1
1
1
0
1
1
1
T1
F1
T2
F2
Positive and Negative Logic (The meaning of High and Low)
1
Signal High
0
Signal Low
Positive Logic
0
Signal High
1
Signal Low
Negative Logic
x
y
F
0
0
0
0
1
0
1
0
0
1 1
1
x
y
F
1
1
1
1
0
1
0 1
1
0 0
0
Note signals are not defined as positive or negative—but as high and low values
Convert from positive to negative logic 0 becomes 1 and 1 becomes 0 = new gate behavior
Book uses positive logic
Integrated Circuits
Logic gates are available as integrated circuits IC or chips. Each IC gas a number
printed on it Identifying its function. Vendors provide data sheets that identify the pin
configurations and IC characteristics.
Texas Instruments
Levels of integration – complexity and density of gates in single package
Small-scale integration (SSI): Fewer than 10 gates, limited by number of pins on
IC package
Medium-scale integration (MSI): Typically 10 to 1000 gates, elementary digital
operations, such as decoders, adders, counters ….
Large-scale integration (LSI): Contain thousands of gates, digital systems such as
processors, memory chips, programmable logic ……..
Very large-scale integration (VLSI): Contain hundreds of thousands of gates,
computer systems, large scale memory arrays, microcomputer chips ….
What you should know
1.
2.
3.
4.
5.
6.
Be able to generate expressions for Sum of Products
and Products of Sums, and draw circuit
Boolean operations
Convert between Canonical forms
The 8 listed gates, schematic symbols, and truth tables
The difference between positive and negative logic
Be familiar with levels of integration and common
characteristics listed in manufactures data sheets.