17 can be expressed in its product of sums form as f2

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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 product-of-sums form as: F1 = (x + y + z) • (x + y + z) • (x + y + z) • (x + y + z) • (x + y + z) or F1 =M0-M2-M3-M5-M6 Similarly, it may be easily verified that the function F 2 of Figure 6.17 can be expressed in its product-of-sums 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 product-of-sums 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 product-of-maxterms (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 +&quot;-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 sum-of-products and the product-of-sums form of Boolean expressions are known as standard forms. One prime reason for liking the sum-of-products or the product-of-sums expressions is their...
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