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Unformatted text preview: decimal equivalent of the binary number of the minterm designated.
In a similar fashion, n variables forming an OR term, with each variable being primed or
unprimed, provide 2n possible combinations called maxterms or standard sums.
The eight maxterms for three variables, together with their symbolic designation, are
listed in Figure 6.16. Any 2n maxterms for n variables may be determined similarly. Each
maxterm is obtained from an OR term of the n variables, with each variable being
unprimed if the corresponding bit is a 0 and primed if it is a 1.
Note that each maxterm is the complement of its corresponding minterm and viceversa.
SumofProducts
A sumofproducts expression is a product term (minterm) or several product terms
(minterms) logically added (ORed) together. For example, the expression xy + xy is a
sumofproducts expression. The following are all sumofproducts expressions:
x x+y
x+y•z
x•y+z
x•y+x•y•z
The following steps are followed to express a Boolean function in its sumofproducts
form:
1. Construct a truth table for the given Boolean function.
2. Form a minterm for each combination of the variables which produces a 1 in the
function.
3. The desired expression is the sum (OR) of all the minterms obtained in step 2.
For example, in case of function Fi of Figure 6.17, the following three combinations of
the variables produce a 1:
001, 100,
and
111
Their corresponding minterms are
xyz,
xyz, and xyz
Hence, taking the sum (OR) of all these minterms, the function Fi can be expressed in its
sumofproducts form
as:
F1 = xyz + xyz + xyz or
F 2 = m , + m4 + m 7
Similarly, it may be easily verified that the function F 2 of Figure 6.17 can be expressed in
its sumofproducts form as:
F2 = xyz + xyz + xyz+xyz
X
y
z
0
0
0
0
1
1
1
1 0
0
l
I 0
0
1
1 0
1
0
1
0
1
0
1 or F2 = m3 + m5 + m6 + m7
F1
F2
0
1
0
0
1
0
0
1 0
0
0
1
0
1
1
1 Figure 6.17. Truth table for functions F1and F2.
It is sometimes convenient to express a Boolean function in its sumofproducts form. If
not in this form, it can...
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 Spring '14

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