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Unformatted text preview: HWK #4 Solution Problem 4.1 (a) h(A,B,C,D)= OR(m 1 ,m 2 ,m 6 ,m 8 ,m 13 ,m 14 ,m 15 ,d ,d 4 ,d 7 ,d 10 ) h(A,B,C,D)= AND(M 3 ,M 5 ,M 9 ,M 11 ,M 12 ,d ,d 4 ,d 7 ,d 10 ) (b) There are 2 minimal SOP expressions. I shall explain first how I obtained these; their Kmaps are provided below. • Looking for a minimal SOP expression, I first observed that only the PI ACD contains m 13 ,only PI A'B'C' contains m 1 , and only PI B'D' contains m 8 . So any minimal SOP expression will contain: B'D' + ACD + A'B'C' • Minterm m 6 and m 14 is covered by both BC' and CD'. Both use a single don't care term but both are loops of size 4 squares, so they will both have a product of two literals, and the minimal sop can be represented with either one of these loops along with the essential PI. • To summarize: the min SOP expressions are as follows: B'D' + ABD + A'B'C' + { BC or CD' } h(A,B,C,D)= B'D' + ABD + A'B'C' + CD' h(A,B,C,D)= B'D' + ABD + A'B'C' + BC There are 4 minimal POS expressions: (B' + C + D) (A' + B + D') { (A + C' + D') or (B + C' + D') { (A + B' + C) or (A + B' + D') } h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + C) (A + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + C) (B + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + D') (A + C' + D') h(A,B,C,D)= (B' + C + D) (A' + B + D') (A + B' + D') (B + C' + D') (c) Your two expressions in part (b) are equal iff every square covered in your SOP K map is uncovered in your POS Kmap. Why? In the SOP Kmap: for every minterm inside a loop, the function is defined to be 1, and for every minterm not inside any loop the function is defined to be 0. The opposite is true in the POS Kmap: for every maxterm inside a loop the function is defined to be 0, and for every maxterm not inside a loop the function is defined to be 1. For this example: • The first SOP expression is equal to the first, third, and fourth POS expressions. • The second POS expression is equal to the second SOP expression. See parts (d) and (e) for additional explanation. (d) There are 2 minimal SOP expressions  see part (b). The function has 4 don't cares. Both of the minimal SOP Kmaps have: d =1, d 10 =1, and d 4 =0. But the values of d 7 differ: • For the first map: d 7 =0 • For the second map: d 7 =1 Hence, the minimal SOP expressions are not equal  because the same squares are not circled in both cases....
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 Fall '08
 BROWN
 Karnaugh map, minimal SOP, minimal SOP expressions, minimal SOP expression, A'B'C

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