3 find the complement of the functions fj and f 2 of

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Unformatted text preview: x-y-z)-(x-y-z) = (x + y + z) ■ (x + y + z) (b) F2=x-(y-z + y x + (y-z)-(y-z) A simpler procedure for deriving the complement of a function is to take the dual of the function and then complement each literal. This method follows from the generalized De Morgan's theorems. Remember that the dual of a function is obtained by interchanging OR and AND operators and O's and l's. The method is illustrated below with the help of an example. Example 6.3. Find the complement of the functions Fj and F 2 of Example 6.2 by taking their dual and complementing each literal. Solution: (a) F, = xyz + xy-z The dual of F1 is: (x + y + z;) ■ (x + y + z) Complementing each literal we get F2 = (x + y + z)-(x (b) F2 = x-(y-z + y Thedualof F2 is: x + (y + z)-(y + z) Complementing each literal we get + z)-(y.(y+z) Canonical Forms for Boolean Functions Minterms and Maxterms A binary variable may appear either in its normal form (x) or in its complement form (x). Now consider two binary variables x and y combined with an AND operator. Since each variable may appear in either form, there are four possible combinations: x-y, x-y, x-y, x-y Each of these four AND terms is called a minterm or a standard product. In a similar manner, n variables can be combined to form 2 n minterms. The 2n different minterms may be determined by a method similar to the one shown in Figure 6.16 for three variables. The binary numbers from 0 to 2 n - 1 are listed under the n variables. Each minterm is obtained from an AND term of the n variables, with each variable being primed if the corresponding bit of the binary number is 0 and unprimed if it is a 1. Variables Minterms Maxterms X y z Term Designation Term Designation 0 0 0 x-y-z m0 x+y+z Mo 0 0 0 1 m, m2 x+y+z x+y+z M, M2 0 1 1 x-y-z 1*1 x-y-z 0 1 x-y-z m3 - x+y+z M3 1 1 0 0 0 1 x-y-z x-y-z m4 m5 x+y+z x+y+z M4 M5 1 1 0 x-y-z m x+y+z M6 1 1 1 x-y-z x+y+z M7 6 m7 Figure 6.16. Minterms and Maxterms for three variables. A symbol for each minterm is also shown in the figure and is of the form nij, where j denotes the...
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