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K-map: x'yz + x'y'z + xyz with gates

Short answer

F(x,y,z) = X'YZ + X'Y'Z + XYZ simplifies to z(x' + y). K-map grouping gives x'z + yz = z(x'+y). Realization: a NOT gate inverts x, an OR gate combines x' and y, and an AND gate takes the OR output and z to produce F.

Simplify the expression using K-Map, F(x,y,z) = X'YZ + X'Y'Z + XYZ and realize it using logic gates.

F(x,y,z) = x'yz + x'y'z + xyz = m3 + m1 + m7 = Σm(1, 3, 7).

K-map

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

Groups:

  • Pair (m1, m3): x = 0, z = 1, y varies gives x'z.
  • Pair (m3, m7): y = 1, z = 1, x varies gives yz.

F = x'z + yz = z(x' + y)

Realization using logic gates

 x ---[NOT]---+
              +--[OR]----+
 y -----------+          +--[AND]---- F
 z -----------------------+

The circuit needs one NOT, one OR and one AND gate. (The SOP form x'z + yz needs one NOT gate, two AND gates and one OR gate.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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