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Chapter%204%20ECE%20201%20Problems[1]

# Chapter%204%20ECE%20201%20Problems[1] - Chapter 4 Problem...

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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) 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 0 0 1 1 0 1 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 0 1 1 1 0 1 1 1 0 1 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 1 1 1 0 1 1 1 1 0 1 1 1 0 1 1 1 1 0 0 1 1 0 0 1 0 0 (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)
- 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 - z w - 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 - .
- 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.
- 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.
- 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 - )
- 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)
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