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K-map: two 3-variable functions

Short answer

i) F = Σ(0,1,3,4,7) simplifies to x'y' + y'z' + yz. ii) F = Σ(0,2,5,6) simplifies to x'z' + yz' + xy'z.

Simplify the following expression: i) F(x,y,z) = xyz + x'y'z + xy'z' + x'y'z' + x'yz ii) F(x,y,z) = Σ(0,2,5,6)

i) F = xyz + x'y'z + xy'z' + x'y'z' + x'yz

Minterms: xyz = 7, x'y'z = 1, xy'z' = 4, x'y'z' = 0, x'yz = 3. So F = Σm(0, 1, 3, 4, 7).

          yz
        00  01  11  10
 x = 0 | 1 | 1 | 1 | 0 |
 x = 1 | 1 | 0 | 1 | 0 |

Groups:

  • (m0, m1): x'y'
  • (m0, m4): y'z' (same column, x varies)
  • (m3, m7): yz

F = x'y' + y'z' + yz (an equally simple form is x'z + y'z' + yz; m4 can only be covered by y'z').

ii) F = Σm(0, 2, 5, 6)

          yz
        00  01  11  10
 x = 0 | 1 | 0 | 0 | 1 |
 x = 1 | 0 | 1 | 0 | 1 |

Groups:

  • (m0, m2): x = 0, z = 0 gives x'z'
  • (m2, m6): y = 1, z = 0 gives yz'
  • m5 alone: xy'z

F = x'z' + yz' + xy'z

✓ Answer checked by Claude Opus 5.5 (high effort)

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