ECE 301 – Digital Electronics
Karnaugh Mapsand
Determining a Minimal Cover
(Lecture #8)
The slides included herein were taken from the materials accompanying
Fundamentals of Logic Design, 6th Edition, by Roth and Kinney,
and were used with permission from Cengage Learning.
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Four-variable K-maprow # A B C D minterm
0 0 0 0 0 m0
1 0 0 0 1 m1
2 0 0 1 0 m2
3 0 0 1 1 m3
4 0 1 0 0 m4
5 0 1 0 1 m5
… …
11 1 0 1 1 m11
12 1 1 0 0 m12
13 1 1 0 1 m13
14 1 1 1 0 m14
15 1 1 1 1 m15
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Minimization: Example #7
Use a Karnaugh map to determine the minimum POS expression
For the following logic function:
F(A,B,C,D) = Σ m(0,1,3,4,5,7,8,11,14)
Specify the equivalent maxterm expansion.
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Minimization: Example #8
Use a Karnaugh map to determine the minimum SOP expression
For the following logic function:
F(A,B,C,D) = Π M(0,2,5,7,8,11,13,15)
Specify the equivalent minterm expansion.
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Minimization: Example #9
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C,D) = Π M(0,1,2,3,6,11,14)
What is the cost of each logic circuit?
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Karnaugh Maps
Karnaugh maps can also be used to minimize incompletely specified functions.
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Minimization: Example #10
Use a Karnaugh map to determine the
1. minimum SOP expression2. minimum POS expression
For the following logic function:
F(A,B,C) = Σ m(4,7) + Σ d(1,3)
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Minimization: Example #11
Use a Karnaugh map to determine theminimum SOP expression
For the following logic function:
F(A,B,C,D) = Π M(0,2,5,6,8,13,15) . Π D(3,4,10)
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Minimization: Example #12
Use a Karnaugh map to determine theminimum POS expression
For the following logic function:
F(A,B,C,D) = Σ m(0,1,2,4,6,8,9,10) + Σ d(3,7,11,13,14)
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Literals and Implicants
● Literal
– Each occurrence of a variable or its complement in an expression
● Implicant (SOP) ← represents a product term
– A single 1 in the K-map
– A group of adjacent 1's in the K-map
● Implicant (POS) ← represents a sum term
– A single 0 in the K-map
– A group of adjacent 0's in the K-map
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Prime Implicants
● Prime Implicant (SOP)
– A product term implicant that cannot be combined with another product term implicant to eliminate a literal.
● Prime Implicant (POS)
– A sum term implicant that cannot be combined with another sum term implicant to eliminate a literal.
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Implicant
Prime
Implicant
Prime
Implicant
Implicant
Implicant
Prime
Implicant
Implicants and Prime Implicants
Additional Prime Implicants?
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If a minterm is covered by only one prime implicant, that
prime implicant is said to be essential, and must be included
in the minimum sum of products (SOP).
Essential
Prime
Implicants
Prime
Implicants
Implicants
Essential Prime Implicants
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Note: 1’s
shaded in blue
are covered by
only one prime
implicant. All
other 1’s are
covered by at
least two prime
implicants.
Identifying Essential Prime Implicants
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Determining a Minimal Cover1. Identify all prime implicants
2. Select all essential prime implicants
3. Select prime implicant(s) to cover remaining terms by considering all possibilities
– Sometimes selection is obvious
– Sometimes “guess” next prime implicant
● Continue, perhaps recursively
● Try all possible “guesses”
4. Determine the Boolean expression
– May not be unique
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Shaded 1’s are
covered by only one
prime implicant.
Essential prime
implicants:
A′B, AB′D′
Then AC′D covers the
remaining 1’s.
Determining a Minimal Cover
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A Minimal Cover
Thus …
A minimal cover is an expression that consists of the fewest product terms (for a SOP expression)
or sum terms (for a POS expression) and the fewest literals in each term.
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