The document describes the process of minimizing Boolean expressions using K-maps. It involves grouping adjacent 1s on the K-map to form product terms. Each group represents a product term in the simplified sum of products (SOP) expression. The goal is to minimize the number of product terms by forming the largest possible groups. The document provides examples of K-maps and the resulting minimized SOP expressions. It also discusses how to map truth tables directly to K-maps and handle "don't care" conditions.

1. K-map Simplification of SOP
Expressions
• Steps for minimisation
– Group the 1s
– Determine the product term for each group
– Sum the resulting product terms
– A minimized SOP results into fewest possible
terms with fewest possible variables
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2. K-map Simplification of SOP
Expressions
• Grouping the 1s
– Group by enclosing adjacent cells containing 1s
– The goal is to maximize the size of the groups and to minimize
the number of groups
– A group must contain either 1,2,4,8 or 16 cells
– Each cell in a group must be adjacent to one or more cells in the
same group, but all cells in the group do not have to be adjacent
to each other
– Always include the largest possible number of 1s in a group in
accordance with the first rule above
– Each 1 on the map must be included in at least one group. The
1s already in a group can be included in another group as long
as the overlapping groups include non-common 1s
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3. K-map Simplification of SOP
Expressions
• Examples
C
0
1
C
0
1
00
1
1
1
01
1
1
11
AB
00
AB
1
01
11
1
10
A’B’C’ + BC + AB
10
1
1
1
A’C’ + AC + B’
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4. K-map Simplification of SOP
Expressions
• Examples
CD
CD
AB
00
01
00
1
1
01
1
1
11
11
10
1
1
AB’D + A’B + A’C’
00
1
01
1
1
1
11
1
1
1
10
1
AB
00
1
10
01
11
10
1
1
1
1
AB’C + BC’ + D’
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5. SOP Determination from the Map
• Rules
• Each group of cells containing 1s creates one
product term composed of all variables that occur
in only one form within the group.
• Variables that occur complemented and
uncomplemented are eliminated
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6. SOP Determination from the Map
• For a 3-variable map
– A 1-cell group yields a 3-variable product term
(e.g. ABC)
– A 2-cell group yields a 2-variable product term
(e.g. AB)
– A 4-cell group yields a 1-variable term (e.g. A)
– An 8-cell group yields a value of 1 for the
expression
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7. SOP Determination from the Map
• For a 4-variable map
– A 1-cell group yields a 4-variable product term (e.g.
ABCD)
– A 2-cell group yields a 3-variable product term (e.g.
ABC)
– A 4-cell group yields a 2-variable term (e.g. AB)
– An 8-cell group yields a 1-variable term (e.g. A)
– A 16-cell group yields a value of 1 for the expression
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8. SOP Determination from the Map
• More examples
CD
AB
00
01
10
1
00
11
1
01
1
1
1
1
11
1
1
1
1
10
1
1
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9. SOP Determination from the Map
• Add a 1 at cell 1010 and determine the
new product terms
• Minimise the following expressions
– AB’C + A’BC + A’B’C + A’B’C’ + AB’C’
– XY’Z + XYZ’ + X’YZ + X’YZ’ + XY’Z’ + XYZ
– AB’C’D’ + A’B’C’D’ + A’BC’D’ + ABC’D’ +
A’B’CD + AB’CD + A’B’CD’ + A’BCD’ + ABCD’
+ AB’CD’
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10. Mapping K-maps Directly from
Truth Tables
Inputs
Output
A B
C
X
0
0
0
0
0
0
1
1
01
0
1
0
0
11
C
1
1
00
0
0
0
1
1
0
0
1
1
1
1
0
1
1
1
1
1
0
1
1
AB
10
0
0
1
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11. Don’t Care Conditions
• Sometimes, some input variable
combinations are not allowed
• BCD – Binary Coded Decimal
– A way to express each of the decimal digits
(0-9) with a binary code of 4 bits
• In BCD code (Chapter 2), there are 6
invalid combinations
– 1010, 1011, 1100, 1101, 1110 and 1111
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12. Don’t Care Conditions
• Don’t care values
BCD Codes
0000
1
CD
0
0001
AB
00
01
11
10
2
0010
00
0
1
3
2
3
0011
01
4
5
7
6
4
0100
11
12
13
15
14
5
0101
10
8
9
11
10
6
0110
7
0111
8
1000
9
1001
No BCD codes for 2digit decimal numbers
(the don’t cares)
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13. Digital System Application
• Introducing the first laboratory work
– The Seven Segment Display (SSD)
– Used in
• Automobile instruments
• Tablet counting control system, etc
– Uses logic circuits that decode a binary coded
decimal (BCD) number and activates the
appropriate digits on the display
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14. The Seven Segment Display
Seven segments:
a, b, c, d, e, f and g
E.g. check your digital
watch, calculator, (some)
phone display digits, etc
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15. Common SSDs
• LED Displays
– Consists of light-emitting diodes (LED)
– Each segment is an LED that emits light when
there is current through it
• LCD Displays
– Liquid Crystal Display
– Operates by polarizing light so that a nonactivated segment reflects incident light and
thus appears invisible against its background
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16. Segment Decoding Logic
• Each segment is used for various decimal
digits (i.e. 0 to 9)
• Each segment must be activated by its
own decoding circuit
• The circuit detects the occurrence of any
of the numbers in which the segment is
used
• Segments that are required to be activated
for each digit (0-9) are:
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17. Segment Decoding Logic
Digit Segments Activated
0
a, b, c, d, e, f
1
b, c
2
a, b, d, e, g
3
a, b, c, d, g
4
b, c, f, g
5
a, c, d, f, g
6
a, c, d, e, f, g
7
a, b, c
8
a, b, c, d, e, f, g
9
a, b, c, d, f, g
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18. Truth Table for the Segment
Logic
• Requires 4 BCD inputs and 7 outputs, one
for each segment in the display (a to g)
• Inputs: A, B, C and D
– A – the least significant bit
– D – the most significant bit
– DCBA (as opposed to ABCD)
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19. Truth Table
Decimal
Digit
Inputs
Segment
Outputs
D C B A
a b c d e f
g
0
0 0
0 0
1 1 1 1 1 1
0
1
0 0
0 1
0
2
0 0
1 0
1
3
0 0
1 1
4
0 1
5
Digit Segments Activated
0
a, b, c, d, e, f
1
b, c
1
2
a, b, d, e, g
0 0
0
3
a, b, c, d, g
0 1
0 1
1
6
0 1
1 0
1
4
b, c, f, g
7
0 1
1 0
1
5
a, c, d, f, g
8
1 0
0 1
1
6
a, c, d, e, f, g
9
1 0
0 0
1
10
1 0
1 0
X
7
a, b, c
11
1 0
1 1
X
8
a, b, c, d, e, f, g
12
1 1
0 0
X
9
a, b, c, d, f, g
13
1 1
0 1
X
14
1 1
1 0
X
15
1 1
1 1
X
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20. Boolean Expressions for the
Segment Logic
• From the truth table, an SOP is developed for
each of the segments
• E.g. SOP for segment ‘a’ is
– a = D’C’B’A’ + D’C’BA’ + D’C’BA + D’CB’A + D’CBA’ +
D’CBA + DC’B’A’ + DC’B’A
• Each product term represents each of the BCD
inputs that activate that segment, therefore
– implementing segment ‘a’ would require AND-OR
circuit containing 8, 4-input AND gates and 1, 8-input
OR gate
– inverters are also required to produce the
complement of each variable
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21. K-Map Minimisation for the
Segment Logic
• SOP for segment ‘a’
• Don’t care’s are used in forming
BA
groups
DC
00
00
1
11
10
1
1
1
01
01
1
1
11
X
X
X
X
10
1
1
X
• a = B + D + CA + C’A’
X
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22. Laboratory Work
• Obtain SOP for the remaining segments
– b, c, d, e, f and g
• For each segment, develop a K-Map and obtain a
simplified expression for the segment
• Draw a circuit for each segment
• There will be duplicate terms across SOP expressions
for the segments – identify them
• For the final circuit, omit all but one duplicates, and
draw the circuit using Deeds software in the laboratory
• Prepare the laboratory report as instructed in the
laboratory handout (report format)
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23. • End of lecture
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