Chapter 4 - Chapter 4 Problem Solutions 4.1. w 0 0 0 0 0 0...

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- 4.1 - Chapter 4 Problem Solutions 4.1. w x y z f (a) (b) (c) (d) (e) (f) (g) (h) 00000 00000110 00010 01000110 00100 00110 01000010 01000 00100110 01011 01100111 01100 01101110 01110 01100010 10001 10100110 10011 01110110 10101 11100110 10111 11001 11010 01100100 11101 11101110 11110 (a) wz - is an implicant since it implies f. However, neither the term w nor the term z - are implicants. Therefore, wz - is a prime implicant (1). (b) y+z is neither an implicant or implicate (6). (c) w+x is an implicate since it is implied by f. However, nether the term w nor the term x are implicates. Therefore, w+x is a prime implicate (3).
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- 4.2 - 4.1. (continued) (d) wx - z is an implicant since it implies f. However, wx - z subsumes wx - which is also an implicant. Therefore, wx - z is an implicant, but not prime (2). (e) xyz - is neither an implicant or implicate (6). (f) w - +y - +z - is an implicate since it is implied by f. However, w - +y - +z - subsumes w+y - which is also an implicate. Therefore, w - +y - +z - is an implicate, but not prime (4). (g) w - +x - +z - is an implicate since it is implied by f. Furthermore, none of the sum terms w - +x - , w - +z - , and x - +z - are implicates. Therefore, w - +x - +z - is a prime implicate (3). (h) w - xy - z is an implicant since it implies f. However, none of the product terms w - xy - , w - xz, w - y - z, and xy - z are implicants. Therefore, w - xy - z is a prime implicant (1). 4.2. (a) (b)
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- 4.3 - 4.2. (continued) (c) (d) (e) (f) 4.3. The Karnaugh map is
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- 4.4 - 4.3. (continued) The implicants correspond to all possible subcubes of 1- cells. one-cell subcubes (minterms) two-cell subcubes four-cell subcubes w - x - y - z - w - x - y - wy w - x - y - zw - x - z - w - x - yz - w - y - z w - xy - z wxy wx - yz - wx - y wx - yz x - yz - wxyz - wyz wxyz wyz - The prime implicants correspond to those subcubes which are not properly contained in some other subcube of 1-cells. The prime implicants are shown on the map. They are wy, w - x - y - , w - x - z - , w - y - z, and x - yz - .
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- 4.5 - 4.4. (a) The prime implicants are xz, xy, y - z, yz - , x - z - , and x - y - . There are no essential prime implicants. (b) The prime implicants are w - x, xz, w - y - z, and wyz. All four prime implicants are essential as a result of the essential 1-cells indicated by asterisks.
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- 4.6 - 4.4. (continued) (c) The prime implicants are w - x - , w - z __, x - z - __, and x - y __. The essential prime implicants are underlined. (d) The prime implicants are y - z - , xy - __, x - z - __, and w - xz ___. The essential prime implicants are underlined.
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- 4.7 - 4.5. (a) The prime implicates are x - +y+z and x+y - +z - . Both prime implicates are essential. (b) The prime implicates are w - +z, x+z, w+x+y - , and w - +x+y. All three prime implicates are essential.
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- 4.8 - 4.5. (continued) (c) The prime implicates are w - +x - , x - +z, and w - +y+z - . All three prime implicates are essential. (d) The prime implicates are x+z - ___, x - +y - +z _____, w - +x - +y - , and w - +y - +z - . The essential prime implicates are underlined.
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- 4.9 - 4.6. (a) Minimal sum: f = x + z Minimal product: f = x + z (b) Minimal sums: f = xy - + x - y + xz f = xy - + x - y + yz Minimal product: f = (x+y)(x - +y - +z)
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- 4.10 - 4.6. (continued) (c) Minimal sum: f = x - y - + x - z + y - z + xyz - Minimal product: f = (x+y - +z)(x - +y+z)(x - +y - +z - ) (d) Minimal sum: f = y - z - + x - yz Minimal products: f = (y+z - )(y - +z)(x - +z - ) f = (y+z - )(y - +z)(x - +y - )
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- 4.11 - 4.7. (a) Minimal sum: f = x + yz - Minimal product f = (x+y)(x+z - ) (b) Minimal sum: f = x - + y + z - Minimal product: f = x - + y + z -
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- 4.12 - 4.7. (continued) (c) Minimal sum: f = x - z - + yz Minimal product: f = (y+z - )(x - +z) (d) Minimal sum: f = y + x - z - Minimal product: f = (x - +y)(y+z - )
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- 4.13 - 4.8. (a) Minimal sum: f = xy + w - x - y - + x - y - z - Minimal products: f = (x - +y)(x+y - )(w - +y+z - ) f = (x - +y)(x+y - )(w - +x+z - ) (b) Minimal sum: f = xz - + wz + x - yz Minimal products: f = (x+z)(w+x - +z - )(w+x+y) f = (x+z)(w+x - +z - )(w+y+z - )
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- 4.14 -
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This note was uploaded on 10/26/2010 for the course ENEE 244 taught by Professor Petrov during the Spring '08 term at Maryland.

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Chapter 4 - Chapter 4 Problem Solutions 4.1. w 0 0 0 0 0 0...

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