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LOGO DKT 122/3
DIGITAL SYSTEM 1
BOOLEAN ALGEBRA (PART 2)
Boolean Algebra
Contents
Boolean Operations & Expression
Laws & Rules of Boolean algebra
DeMorgan’s Theorems
Boolean analysis of logic circuits
Simplification using Boolean Algebra
Standard forms of Boolean Expressions
Boolean Expressions & truth tables
The Karnaugh Map
Karnaugh Map SOP minimization
Karnaugh Map POS minimization
Programmable Logic
Boolean Algebra (Cont.)
Solve this..
B
A
X
(a) Write the equivalent Boolean expression of circuit shown above.
(b) Simplify the Boolean expression found in (a).
C
Solve this..
� Simplify the following Boolean expressions:
� (AB(C + BD) + AB)C
� ABC + ABC + ABC + ABC + ABC
� Write the Boolean expression of the following circuit.circuit.
RULE:Break the longest
bar first!
RULE:Break the longest
bar first!
� Sum-of-products (SOP)
Refer to two or more product terms, summed (added) by Boolean addition
Standard Forms of Boolean Expressions
summed (added) by Boolean addition
� Product-of-sums (POS)
Refer to multiplication of two or more sum terms
Sum-of-products (SOP)
� In an SOP expression, a single overbarcannot extend over more than one variable
The standard SOP form
Refer to expression in which all the variables in the domain appear in each product term in the expression
Sum-of-products (SOP)
expression
Example:
ABCD + ABCD + ABCD (standard SOP expression)
ABC + ABD + ABCD (non-standard SOP expression)
� All variables appear in each product term.� Each of the product term in the expression is
called as minterm.
Example BCACABCBACBAf ++=),,(
Sum-of-products (SOP)
The standard SOP form
Example BCACABCBACBAf ++=),,(
In compact form, f(A,B,C) may be written as
632),,( mmmCBAf ++=
)6,3,2(),,( mCBAf Σ=
Product-of-sums (POS)
� In a POS expression, a single overbarcannot extend over more than one variable
Refer to expression in which all the variables in the domain appear in each sum term in the expression
Example:
Standard Forms of Boolean Expressions
The standard POS form
Example:
(A + B + C)(B + C + D)(A + B + C + D) (non-standard POS expression)
(A + B + C + D)(A + B + C + D) (standard POS expression)
• All variables appear in each product term.• Each of the product term in the expression is called as maxterm.
Example: )()()(),,( CBACBACBACBAf ++•++•++=
Standard Forms of Boolean Expressions
The standard POS form
Example: )()()(),,( CBACBACBACBAf ++•++•++=
In compact form, f(A,B,C) may be written as
541),,( MMMCBAf =
)5,4,1(),,( MCBAf π=
Identify each of the following expressions as SOP, standard SOP, POS or standard POS:
(i) AB + ABD + ACD(ii) (A + B + C)(A + B + C)(iii) ABC + ABC
Solve this..
(iii) ABC + ABC(iv) A(A + C)(A + B)
Convert the following Boolean expressions to SOP form:
(i) (A + B)(B + C + D)(ii) AB + B(CD + EF)
Convert the following Boolean expressions into standard
SOP form:
AB + ABCD
Solution
Example (Standard SOP)
Solution
1st step: AB = AB (C + C) = ABC + ABC
2nd step: ABC (D + D) + ABC (D + D)
= ABCD + ABCD + ABCD + ABCD
3rd step: ABCD + ABCD + ABCD + ABCD + ABCD
Convert the following Boolean expressions into standard
POS form:
(A + B + C)(B + C + D)
Solution
Example (Standard POS)
Solution
1st step: A + B + C = A + B + C + DD
= (A + B + C + D) (A + B + C + D)
2nd step: B + C + D = B + C + D + AA
= (A + B + C + D)(A + B + C + D)
Final answer: Combine answer for 1st and 2nd step
Boolean Expressions & Truth Tables
Converting SOP to Truth Table
� Examine each of the products to determine where the product is equal to a 1.
� Set the remaining row outputs to 0.
Converting POS to Truth Table
�Opposite process from the SOP expressions.
� Each sum term results in a 0.
� Set the remaining row outputs to 1.
Boolean Expressions & Truth Tables
Determining Standard Expressions from Truth Table
INPUT OUTPUT
A B C X
0 0 0 0
SOP Expressions
X = ABC + ABC + ABC +
Boolean Expressions & Truth Tables
0 0 0 0
0 0 1 0
0 1 0 0
0 1 1 1
1 0 0 1
1 0 1 0
1 1 0 1
1 1 1 1
X = ABC + ABC + ABC + ABC
POS Expressions
X = (A + B + C)(A + B + C)(A + B + C)(A + B + C)
Solve this..
� Convert the following SOP expression to an equivalent POS expression:
CBACBACABABCCBAf +++=),,(
� Develop a truth table for the expression:
)()()()(),,( CBACBACBACBACBAf ++•++•++•++=
CBACBACABABCCBAf +++=),,(
The Karnaugh Map (K-Map)
� Karnaugh Map (K-map) is an array of cells in which each cell represents a binary value of the input variables
� K-Mapping is used to minimize the number of logic gates that are required in a digital circuit. This will replace Boolean reduction when the This will replace Boolean reduction when the circuit is large.
� The number of cells in a K-map is equal to the total number of possible input variable combinations
� K-Map is similar to Truth Table because it present all possible values of input variables and the resulting output for each value
� The map is made up of a table of every possible SOP using the number of variables that are being used.
If 2 variables are used, then a 2X2 map is used,
The K-map
is used,
If 3 variables are used, then a 4X2 map is used,
If 4 variables are used, then a 4X4 map is used,
If 5 variables are used, then a 8X4 map is used
� Mapping Standard SOP Expression
For an SOP in standard form, a 1 is placed on the K-map for each product term in the expression
K-map SOP Minimization
Gray Code
Example of standard SOP form
� Mapping Non-standard SOP ExpressionExpand the non-standard expression
A + AB + ABC
K-map SOP Minimization
A + AB + ABC
QuestionHow to expand this expression?
QuestionHow to expand this expression?
● Mapping directly from a truth table to a K-map
K-map SOP Minimization
Thomas L. Floyd
Digital Fundamentals, 9e
Copyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458
All rights reserved.
Notice that the map is going false to true, left to right and top to bottomA
B B
0 10 1
2 variables K-map
K-map SOP Minimization
The upper right hand cell is A B if X= A B then put an X in that cell
This show the expression true when A = 0 and B = 0
A
A
B B
1
A 2 32 3
If X=AB + AB then put an X in both of these cells
A
A
B B
1
1
2 variables K-map
K-map SOP Minimization
these cells
From Boolean reduction we know that A B + A B = B
From the Karnaugh map we can circle adjacent cell and find that X = B
A
A
B B
1
1
A 1
Gray Code 0 1
C C
K-map SOP Minimization
3 variables K-map
00 A B
01 A B
11 A B
10 A B
0 10 1
2 32 3
6 76 7
4 54 5
0 1
K-map SOP Minimization
X = A B C + A B C + A B C + A B C
3 variables K-map
00 A B
01 A B
11 A B
10 A B
0 1
C C
Each 3 variable term is one cell on a 4 X 2
Karnaugh map
Each 3 variable term is one cell on a 4 X 2
Karnaugh map
1 1
1 1
0 1
One
X = A B C + A B C + A B C + A B C
K-map SOP Minimization
3 variables K-map
00 A B
01 A B
11 A B
10 A B
0 1
C COne simplification could be
X = A B + A B
1 1
1 1
X = A B C + A B C + A B C + A B C
0 1Another
X = A B C + A B C + A B C + A B C
K-map SOP Minimization
3 variables K-map
00 A B
01 A B
11 A B
10 A B
C CAnother simplification could be
X = B C + B C
A Karnaugh Map does wrap around
1 1
1 1
X = A B C + A B C + A B C + A B C
0 1
X = A B C + A B C + A B C + A B C
K-map SOP Minimization
3 variables K-map
00 A B
01 A B
11 A B
10 A B
0 1
C CThe bestsimplification would be
X = B
1 1
1 1
Conclusions
� One cell requires 3 variables
K-map SOP Minimization
3 variables K-map
� One cell requires 3 variables
� Two adjacent cells require 2 variables
� Four adjacent cells require 1 variable
� Eight adjacent cells is a 1
Gray Code 0 0 0 1 1 1 1 0
C D C D C D C D
K-map SOP Minimization
4 variables K-map
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
0 1 3 20 1 3 2
4 5 7 64 5 7 6
12 13 15 1412 13 15 14
8 9 11 108 9 11 10
Gray Code 0 0 0 1 1 1 1 0
C D C D C D C D
Simplify:
X = A B C D + A B C D + A B C D + A B C D + A B C D + A B C D
K-map SOP Minimization
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
1
1
1
1
1
1
X = ABD + ABC + CD
Now, try it with Boolean reductions..
K-map SOP Minimization
Conclusions
� One Cell requires 4 variables
4 variables K-map
� One Cell requires 4 variables
� Two adjacent cells require 3 variables
� Four adjacent cells require 2 variables
� Eight adjacent cells require 1 variable
� Sixteen adjacent cells give a 1 or true
Simplify
Z = B C D + B C D + C D + B C D + A B C
Gray Code 00 01 11 10
C D C D C D C D
K-map SOP Minimization
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
1 1
1 1
1 1
1 1
1
1
Z = BD + C
Example
Simplify the following circuit using K-map method.
Y = A + B + B C + ( A + B ) ( C + D)
Example (Cont.)
Y = A B + B C + A + B + ( C + D )
Y = A B + B C + (A + B ) + C D
Simplified SOP expression
Gray Code 00 01 11 10
C D C D C D C DY = 1
Then, map the SOP expression into the K-map & simplify the equation
Example (Cont.)
00 A B
01 A B
11 A B
10 A B
C D C D C D C D
1 1
1 1
1 1 1 1
1 1 1 1
1 1
1 1
K-map POS Minimization
Assume A, B, C, and D are variables.
3 variables 4 variables
K-map POS Minimization
3 variables K-map
K-map POS Minimization
4 variables K-map
K-map POS Minimization
4 variables K-map
Input Output
K-map Minimization – Don’t Cares
3 variables with output “don’t cares” (X)
K-map Minimization – Don’t Cares
4 variables with output “don’t cares” (X)
Determine the minimal SOP using K-Map:
14,15)D(5,12,13, 9,10)M(0,2,6,8, D)C,B,F(A, π=
Example
D refers to “don’t cares”
What is this?
AB
CD
00
00 01 11 10
0 1 1 00 1 3 2
Example (Cont.)
14,15)D(5,12,13, 9,10)M(0,2,6,8, D)C,B,F(A, π=
00
01
11
10
1 X 1 0
X X X X
0 0 1 0
0 1 3 2
4 5 7 6
12 13 15 14
8 9 11 10
DACBCDDCBAF ++=),,,(
Minimum SOP expression is:
CD
ADBC
Solve this..
Reduce (a), (b) and (c) using K-map:
(a) (b)
∑ ∑+= 13) 12, (6, d 9) 7, 5, 3, (1, m D) C, B, f(A, (c)