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MELJUN CORTES Computer Trends IssuesTRANSCRIPT
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� simply called the K-map
� a systematic method of reducing the complexity
KarnaughMaps
� a systematic method of reducing the complexity of algebraic expressions
� guarantees the simplest Boolean expression using very straightforward procedures
� can be presented in different ways basing on
the number of variables the Boolean function has
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� 22 = 4 minterms
� a map with four quadrants
Two-VariableMap
� a map with four quadrants
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Example 4:
Obtain the K-Map of the function: F = ΣΣΣΣ m(0,3)
Two-VariableMap
Obtain the K-Map of the function: F = ΣΣΣΣ m(0,3)
Solution:
� there are two minterms that has a value of 1(m0, m3)
� put a 1 on the appropriate squares of the two-
Truth Table:
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� put a 1 on the appropriate squares of the two-variable K-map
� blank squares automatically have 0 values (you
may not write the 0 in the squares)
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Example 5:
Obtain the K-Map of the function: F = XZ
Two-VariableMap
Obtain the K-Map of the function: F = XZ
Solution:
Truth Table
� the minterm that has a value of 1 is m3
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� 23 = 8 minterms
� have eight quadrants containing a minterm
Three-VariableMap
� have eight quadrants containing a minterm each
� does not follow the normal binary count sequence like that of the two-variable map
� the way the minterms are assigned to each of
the squares differ by only one variable
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� the fourth column of this map wraps around to the first column
Three-VariableMap
� count from the outside to the middle:
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Example 6:
Obtain the K-Map of the function: F = ΣΣΣΣ
Three-VariableMap
Obtain the K-Map of the function: F = ΣΣΣΣm(1,3,5,7)
Solution:
� this is a three-variable map since its highest value is 7
� four minterms (1,3,5,7) has a value of 1
Truth Table:
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Example 7:
Obtain the K-Map of the function:
Three-VariableMap
Obtain the K-Map of the function:
F = X’YZ’ + X’YZ + XY’Z + XY’Z’
Solution:
Truth Table:
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Example 8:
Obtain the K-Map of the function:
Three-VariableMap
Obtain the K-Map of the function:
F = X + X’Z + Y’Z
Solution:
Truth Table:
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� 24 = 16 minterms
� have sixteen quadrants containing a minterm
Four-VariableMap
� have sixteen quadrants containing a minterm each
� an extension of the three-variable map
� the assignment of columns and rows are similar to that of the three-variable map
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Example 9:
The Boolean function: F = ΣΣΣΣ m(0,3,4,15) needs
Four-VariableMap
The Boolean function: F = ΣΣΣΣ m(0,3,4,15) needs to be represented in a K-Map.
Solution:
Minterms: m0, m3, m4, m15
Truth Table:
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Example 10:
The Boolean function:
Four-VariableMap
The Boolean function:
F (A, B, C, D) = ΣΣΣΣ m(0,1,5,7)
needs to be represented in a K-Map.
Solution:
Minterms: m0, m1, m5, m7
Truth Table:
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� remember: K-map is directly related to the minterms (and therefore, its truth table)
K-Map Simplification
K-Map for F (X, Y) = 1
K-Map for F (X, Y) = ΣΣΣΣ m(0,1,2,3)
or
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or
F (X, Y) = m0+m1+m2+m3 = X’Y’+X’Y+XY’+XY
= X’Y’+X’Y+XY’+XY
= X’(Y’+Y) + X(Y’+Y) Distributive
= X’(1) + X(1) Postulate 6
= 1
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� What happens when we take out one minterm from the equation below?:
K-Map Simplification
F (X, Y) = ΣΣΣΣ m(0,1,2,3)
E (X, Y) = ΣΣΣΣ m(1,2,3)
OR
E (X, Y) = m1+m2+m3 = X’Y+XY’+XY
Simplify using Boolean algebra:
E = X’Y + XY’ + XY
= X’Y + X(Y’+Y) Distributive Prop.
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= X’Y + X(Y’+Y) Distributive Prop.
= X’Y + X(1) Postulate 6
= X’Y + X
= X’Y + (X+XY) Theorem 6
= X’Y + X + XY
= X + Y(X’+X) Distributive Prop.
= X + Y(1) Postulate 6
= X + Y
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Step 1: Draw the K-Map
K-Map Simplification
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Step 2: Group into clusters
� Take note of all the adjacent squares that have
K-Map Simplification
� Take note of all the adjacent squares that have a value of “1”
Rules in Grouping:
1. must be of size 1,2,4,8,16....2n
2. all the 1’s that can be grouped must be included in a cluster of maximum size
3. the minimum size for the cluster is two
4. combine squares that are adjacent
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Step 3: Get the simplified expression from the clustering
K-Map Simplification
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Step 3 (cont.):
TIP: Get only the variables that appear in all
K-Map Simplification
TIP: Get only the variables that appear in all the squares included in the cluster
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� the two values (X,Y) are the terms included for the sum of products expression
E (X, Y) = ΣΣΣΣ m(1,2,3) = X’Y+XY’+XY
E = X + Y
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Example 11:
Draw the K-map of the truth table shown here
K-Map Simplification
Draw the K-map of the truth table shown here and minimize the resulting expression.
Solution:
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Step 1: Draw the K-Map
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Solution (cont.):
K-Map Simplification
Step 2: Group into clusters
Step 3: Write the equivalent expression
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Example 12:
Design a logic circuit to implement the following
K-Map Simplification
Design a logic circuit to implement the following truth table:
Solution:
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Step 1: Draw the K-Map
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Solution (cont.):
Step 2: Group into clusters
K-Map Simplification
Step 2: Group into clusters
Step 3: Write the equivalent expression
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Solution (cont.):
Step 3: Write the equivalent expression
K-Map Simplification
Step 3: Write the equivalent expression
F = A’B’ + BC’ + B’C’
= A’B’ + C’(B + B’) Distributive
= A’B’ + C’(1) Postulate 6
F = A’B’ + C’
The Wrapping-Around Method:
Back to Step 2:Group into clusters - wrap the
leftmost column with the rightmost column to get the
maximum number of groupings
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maximum number of groupings
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Solution (cont.):
Step 3: Write the equivalent expression
K-Map Simplification
Step 3: Write the equivalent expression
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Solution (cont.):
Step 4: Draw the logic diagram
K-Map Simplification
Step 4: Draw the logic diagram
F = A’B’ + C’
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Three- and Four-Variable K-Maps
with Wrap-around
Summary forWrapping Around
with Wrap-around
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Summary forWrapping Around
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Summary forWrapping Around
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Summary forWrapping Around
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Summary forWrapping Around
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Summary forWrapping Around
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Summary forWrapping Around
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Example 13:
Design a logic circuit to implement the following
K-Map Simplification
Design a logic circuit to implement the following truth table:
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Solution:
Step 1: Draw the K-Map
K-Map Simplification
Step 1: Draw the K-Map
Step 2: Group into clusters
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Solution (cont.):
Step 3: Write down the terms and the final
K-Map Simplification
Step 3: Write down the terms and the final expression
upper-left square cluster: B’C’
outer two-four cluster: D’
thus,
F = B’C’ + D’
Step 4: Draw the logic diagram
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