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Unformatted text preview: tep 2.
For example in case of function Fi of Figure 6.17, the following five combinations of the
variables produce a 0:
000, 010,011, 101, and 110
Their corresponding maxterms are:
(x + y + z), (x + y + z), (x + y + z), (x + y + z), and (x + y + z)
Hence, taking the product (AND) of all these maxterms, the function Fi can be expressed
in its productofsums form as:
F1 = (x + y + z) • (x + y + z) • (x + y + z) • (x + y + z) • (x + y + z)
or
F1 =M0M2M3M5M6
Similarly, it may be easily verified that the function F 2 of Figure 6.17 can be expressed in
its productofsums form as:
F2 = (x + y + z)(x + y + z) • (x + y + z) • (x + y + z) or
F2 = M0 • M, • M2 • M4
In order to express a Boolean function in its productofsums form, it must first be
brought into a form of OR terms. This may be done by using the distributive laws:
x + y • z = (x + y) • (x + z)
Then any missing variable (say x) in each OR term is ORed with the form x • x. This
procedure is explained with the following example.
Example 6.5.
Express the Boolean function F = x • y + x • z in the productofmaxterms (sums) form.
Solution:
At first we convert the function into OR terms using the distributive law:
F=x•y+x•z
= (x • y + x) • (x • y + z)
= (x+x) • (y+x) • (x + z) • (y + z) = (x+y) • (x + z) • (y + z)
The function has three variables x, y and z. Each OR term is missing one variable,
therefore:
x +y = x +y + Z • Z = (x +y + Z) • (X +y+ Z)
x + z = x + z +"y • y =(x + z + y) • (x + z+ y) = (x +y + z)(x + y +z) y + z
= x • +y + z = (x + y + z)(x+y + z)
Combining all the terms and removing those that appear more than once, we finally
obtain:
F = (x + y + z) • (x + y + z) • (x + y + z) • (x + y+ z) = M0 • M2 • M4 • M5
A convenient way to express this function is as follows:
F (x, y, z) = II (0, 2, 4, 5)
The product symbol FI denotes the ANDing of maxterms. The numbers following it are
the maxterms of the function.
The sumofproducts and the productofsums form of Boolean expressions are known as
standard forms. One prime reason for liking the sumofproducts or the productofsums
expressions is their...
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 Spring '14

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