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Simplification Techniques
DR. RISANURI HIDAYAT
Sum-of-Products Expressions
• A sum-of-products expression contains the
sum of different terms
• It can be obtained from the truth table
directly by considering those input
combinations that produce a logic ‘1’ at the
output
• A sum-of-products expression is also known as
a minterm expression.
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Sum-of-Products Expressions
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Product-of-Sums Expressions
• A product-of-sums expression contains the
product of different terms
• It can be obtained from the truth table by
considering those input combinations that
produce a logic ‘0’ at the output
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Product-of-Sums Expressions
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Σ and Π notations
• Σ and Π notations are respectively used to represent sum-ofproducts and product-of-sums expressions.
• Sum-of-products given by
• The different terms represent 0001, 0101, 1000, 1001 and 1111.
The decimal equivalent of these terms enclosed in the then gives
the notation for the given Boolean function. That is,
f(A, B, C, D) = Σ 1, 5, 8, 9, 15
f’(A, B, C, D) = Σ 0, 2, 3, 4, 6, 7, 10, 11, 12, 13, 14
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• Let us now take the case of a product-of-sums
Boolean function and its representation in
nomenclature. Let us consider the Boolean
function
• The binary numbers represented by the different
sum terms are 0011, 1011, 1100 and 0111 (3, 11,
12, 7). Therefore,
f(A, B, C, D) = Π 3, 7, 11, 12
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• Consider a function of three variables f(A, B, C):
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• Consider the function
• Consider the following
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• This is because the FIRST RULE of the Tabular method for
two terms to combine, and thus eliminate one variable, is
that they must differ in only one digit position.
• Bear in mind that when two terms are combined, one of
the combined terms has one digit more at logic 1 than the
other combined term. This indicates that the number of 1's
in a term is significant and is referred to as its index.
• For example: f(A, B, C, D)
0000...................Index 0
0010, 1000.............Index 1
1010, 0011, 1001.......Index 2
1110, 1011.............Index 3
1111...................Index 4
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References
• Tabular Method of Minimisation
http://www.ee.surrey.ac.uk/Projects/Labview/mi
nimisation/tabular.html
• K-maps (Karnaugh Maps)
http://www.cc.gatech.edu/~howardz/micellaneo
us/gre_cs_sub/karnaugh_maps.htm
• Minterm vs maxterm solution
http://www.allaboutcircuits.com/vol_4/chpt_8/8.
html
•
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