Professor CK Cheng
CSE Dept.
UC San Diego
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Example
Given F =
S m (3, 5), D =
S m (0, 4) c b
0 2 6 4
0 0 -
1 3 7 5
0 1 0 1 a
Primes:
S m (3),
S m (4, 5)
Essential Primes:
S m (3),
S m (4, 5)
Min exp: f(a,b,c) = a’bc + ab’
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Boolean Expression K-Map
Variable x i and its compliment x i
’
 Two half planes Rx i
, and Rx i
’
Product term P
(
P x i
* e.g. b’c’)

Intersect of Rx i
* for all i in P e.g. Rb’ intersect Rc’
Each minterm 
One element cell
Two minterms are adjacent iff they differ by one and only one variable, eg: abc’d, abc’d’

The two cells are neighbors
Each minterm has n adjacent minterms

Each cell has n neighbors
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Procedure Input: Two sets of F R D
1) Draw K-map.
2) Expand all terms in F to their largest sizes (prime implicants).
3) Choose the essential prime implicants.
4) Try all combinations to find the minimal sum of products. (This is the most difficult step)
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Example
Given F =
S m (0, 1, 2, 8, 14)
D =
S m (9, 10)
1. Draw K-map b
0 4 12 8
1 0 0 1
1 5 13 9
1 0 0 -
3 7 15 11
0 0 0 0 c
2 6 14 10
1 0 1 a d
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2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms.
S m (0, 1, 8, 9),
S m (0, 2, 8, 10),
S m (10, 14)
3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes.
S m (0, 1, 8, 9),
S m (0, 2, 8, 10),
S m (10, 14)
4. Min exp:
S m (0, 1, 8, 9) +
S m (0, 2, 8, 10) +
S m (10, 14) f(a,b,c,d) = b’c’ + b’d’+ acd’
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Another example
Given F =
S m (0, 3, 4, 14, 15)
D =
S m (1, 11, 13)
1. Draw K-map b
0 4 12 8
1 1 0 0
1 5 13 9
0 0
3 7 15 11
1 0 1 c
2 6 14 10
0 0 1 0 a d
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2. Prime Implicants: Largest rectangles that intersect On Set but not Off Set that correspond to product terms.
E.g.
S m (0, 4),
S m (0, 1),
S m (1, 3),
S m (3, 11),
S m (14, 15),
S m (11, 15),
S m (13, 15)
3. Essential Primes: Prime implicants covering elements in F that are not covered by any other primes.
E.g.
S m (0, 4),
S m (14, 15)
4. Min exp:
S m (0, 4),
S m (14, 15), (
S m (3, 11) or
S m (1,3) ) f(a,b,c,d) = a’c’d’+ abc+ b’cd (or a’b’d)
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d c
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
Five variable K-map e d c
16 20 28 24
17 21 29 25
19 23 31 27
18 22 30 26 b b a e
Neighbors of m
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Neighbors of m
10 are: minterms 1, 4, 7, 13, and 21 are: minterms 2, 8, 11, 14, and 26
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e d
0 4 12 8
Six variable K-map d
16 20 28 24
1 5 13 9
3 7 15 11
2 6 14 10 f e
17 21 29 25
19 23 31 27
18 22 30 26 e c d
32 36 44 40
33 37 45 41
35 39 47 43
34 38 46 42 f e c d
48 52 60 56
49 53 61 57
51 55 63 59
50 54 62 58 c c b f f a
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Implicant : A product term that has non-empty intersection with on-set F and does not intersect with off-set R .
Prime Implicant : An implicant that is not covered by any other implicant .
Essential Prime Implicant : A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants .
Implicate : A sum term that has non-empty intersection with off-set
R and does not intersect with on-set F.
Prime Implicate : An implicate that is not covered by any other implicate.
Essential Prime Implicate : A prime implicate that has an element in off-set R but this element is not covered by any other prime implicates .
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Min product of sums
Given F =
S m (3, 5), D =
S m (0, 4) c b
0 2 6 4
0 0 -
1 3 7 5
0 1 0 1 a
Prime Implicates:
P
M (0,1),
P
M (0,2,4,6),
P
M (6,7)
Essential Primes Implicates:
P
M (0,1),
P
M (0,2,4,6),
P
M (6,7)
Min exp: f(a,b,c) = (a+b)(c )(a’+b’)
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a b a’ b’ c
Corresponding Circuit f(a,b,c,d)
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Quiz
Given F =
S m (0, 6), D =
S m (2, 7),
1. Fill the Karnaugh map.
2. Identify all prime implicates
3. Identify all essential primes.
4. Find a minimal expression in product of sums format.
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Another min product of sums example
Given R =
S m (3, 11, 12, 13, 14)
D =
S m (4, 8, 10)
K-map c b
0 4 12 8
1 0 -
1 5 13 9
1 1 0 1
3 7 15 11
0 1 1 0
2 6 14 10
1 1 0 a d
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Prime Implicates:
P
M (3,11),
P
M (12,13),
P
M(10,11),
P
M (4,12),
P
M (8,10,12,14)
Essential Primes:
P
M (8,10,12,14),
P
M (3,11),
P
M(12,13)
Exercise: Derive f(a,b,c,d) in minimal product of sums expression.
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•
Karnaugh Maps: Two dimensional truth table which mimics an n-variable cube with imaginary adjacency.
•
Theme: Relation between Boolean algebra and Karnaugh maps.
•
Key words: Primes, Essential Primes
•
Goal: Minimal expression in the format of sum-of-products or product-of-sums.
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