POST BY : NURUL ILHAM KAMALIAH BT AHMAD B031210353
3.2: COMBINATIONAL
CIRCUITS
v
Alogic block contains no memory and computes the
output given the current inputs.
v
Can be defined in three ways :
1.
Truth table
2.
Graphical symbols
3.
Boolean equations
BOOLEAN EQUATION FORMS
v
A boolean
algebra is the combinations of variables and operators.
v
All Boolean expression can be represented in two
forms:
Ø
Sum-of-products(SOP) = Combination of input values that produce 1s is convert into
equivalentvariables,ANDed
together then ORed with other
combination variable with the same output.
Example :
- The truth table :
F = A’B’+ ABC’D
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A
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B
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C
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D
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F
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0
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0
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0
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0
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1
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0
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0
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0
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1
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1
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0
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0
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1
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0
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1
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0
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0
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1
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1
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1
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0
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1
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0
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0
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0
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0
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1
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0
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1
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0
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0
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1
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1
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0
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0
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0
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1
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1
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1
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0
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1
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0
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0
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0
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0
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1
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0
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0
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1
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0
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1
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0
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1
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0
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0
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1
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0
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1
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1
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0
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1
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1
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0
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0
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0
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1
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1
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0
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1
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1
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1
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1
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1
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0
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0
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1
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1
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1
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1
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0
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SOP
expression :
Ø Product-of-sum(POS) = Input combinations that produce 0s in sum terms
(ORed variables) are ANDed together.
Convert input values that
produce 0s into equivalent variables
ORed the variables,thenANDed
with other ORed forms
Usually use if more 1s produce in output function
EXAMPLE :
F = (A+B+C)(A+B+C’)(A+B+C’)(A’+B+C’)
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A
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B
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C
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F
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0
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0
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0
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0
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0
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0
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1
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0
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0
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1
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0
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0
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0
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1
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1
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1
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1
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0
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0
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1
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1
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0
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1
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0
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1
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1
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0
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1
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1
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1
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1
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1
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POS expression :
F =
(A+B+C)(A+B+C’)(A+B+C’)(A’+B+C’)
SIMPLIFICATION OF BOOLEAN EQUATION
- Two ways to simplify Boolean equation,
1.
Law Of Boolean Algebra – rules to simplify
expression
2.
Karnaugh Map – A grid-like representation of a
truth table
LAW OF BOOLEAN ALGEBRA
Table
3-2: Basic Laws of Boolean Algebra
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AND Form
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OR Form
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Identity Law
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A·1 = A
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A + 0 = A
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Zero and One Law
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A·0
= 0
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A+1 = 1
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Inverse Law
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A·A’ = 0
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A+A’ = 1
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Indempotent Law
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A·A = A
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A+A = A
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Commutative Law
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A·B = B·A
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A+B = B+A
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Associative Law
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A·(B·C) = (A·B)·C
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A+(B+C) = (A+B)+C
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Distributive Law
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A+(B·C) = (A+B)·(A+C)
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A·(B+C)
= (A·B)+(A·C)
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Absorbtion Law
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A(A+B) = A
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A+A·B = A
A+A’B = A+B
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DorMorgan’s Law
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(A·B)’ = A’+B’
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(A+B)’ = A’·B’
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Double Complement Law
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=
X = X
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De Morgan’s Law
NOTE:
If you break the line,you change the sign.
(A·B)’ = A’+ B’
(A+B)’ = A’·B’
POST BY : NUR SYAZWAN BT ABDUL MALEK B031210119
3.3 Karnaugh Map
(K-Map)
- K-Map provides a simple method on minimizing Boolean
expressions.
- However, it will be ineffective for more than four inputs.
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For n inputs, there are 2n
cells needed to build the K-Map
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Example
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A
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B
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Minterm
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0
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0
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A’B’
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0
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1
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A’B
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1
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0
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AB’
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1
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1
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AB
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A
B
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A’
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A
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B’
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B
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K-Map for two
variables(A and B)
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A
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B
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C
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Minterm
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0
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0
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0
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A’B’C’
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0
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0
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1
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A’B’C
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0
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1
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0
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A’BC’
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0
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1
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1
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A’BC
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1
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0
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0
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AB’C’
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1
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0
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1
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AB’C
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1
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1
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0
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ABC’
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1
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1
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1
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ABC
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BC
A
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B’C’
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B’C
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BC
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BC’
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A’
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A
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K-Map for three variables(A,B and C)
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A
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B
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C
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D
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Minterm
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0
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0
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0
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0
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A’B’C’D’
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0
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0
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0
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1
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A’B’C’D
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0
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0
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1
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0
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A’B’CD’
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0
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0
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1
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1
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A’B’CD
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0
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1
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0
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0
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A’BC’D’
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0
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1
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0
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1
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A’BC’D
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0
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1
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1
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0
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A’BCD’
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0
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1
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1
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1
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A’BCD
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1
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0
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0
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0
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AB’C’D’
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1
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0
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0
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1
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AB’C’D
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1
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0
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1
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0
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AB’CD’
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1
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0
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1
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1
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AB’CD
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1
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1
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0
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0
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ABC’D’
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1
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1
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0
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1
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ABC’D
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1
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1
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1
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0
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ABCD’
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1
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1
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1
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1
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ABCD
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CD
AB
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C’D’
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C’D
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CD
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CD’
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A’B’
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A’B
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AB
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AB’
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K-Map for four variables (A,B,C, and D)
A product term that includes all of the variables once,
either complemented or not complemented is called a minterm.
For example,
-
If there are two inputs or variables, there are
four minterms(A’B’,A’B,)
-
If there are inputs or variables,there are 8 mintermA’B’C’ , A’B’C , A’BC’ , A’BC , AB’C’ ,
AB’C , ABC’ and ABC.
Example
F = A+B
- The truth table:
|
|
A
|
B
|
A+B
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A’B’
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0
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0
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0
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A’B
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0
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1
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1
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AB’
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1
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0
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1
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AB
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1
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1
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1
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- Karnaugh Map:
|
A
B
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A’
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A
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B’
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0
|
1
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B
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1
|
1
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- Boolean Law:
F = A’B + AB’ + AB
=A’B + AB + AB’ +
AB AB + AB = AB
=B (A’+ A) + A(B’ + B)
=B + A
=A + B
GROUPING 1’s IN KARNAUGH MAP
1- The groups can ONLY contain 1s
2- Only 1s adjacent cells can be grouped ; diagonal
grouping is not allowed.
3- The number of 1s in a group must be a power of
2,means a group can contain 2,4,8, or 16 of
1s.
4- The groups must be as large as possible while still following
all rules.
5- All 1s must belong to a group, even if it is a
group of one
6- Overlapping
groups are allowed.
7- Wrap around is allowed.
8- Use the fewest number of groups possible.
