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K-map with don't cares, NAND only

Short answer

With don't-cares d(3,4,6), K-map groups: row A=0 gives A', columns B=0 gives B'. Simplified F = A' + B'. Using De Morgan, F = (AB)' realized by a single NAND gate.

Simplify the given function using K-map method. F(A,B,C) = Σ(0, 1, 2, 5) + D(3, 4, 6) and implement the simplified circuit using NAND only.

F(A,B,C) = Σm(0, 1, 2, 5) + d(3, 4, 6).

K-map (X = don't care)

          BC
        00  01  11  10
 A = 0 | 1 | 1 | X | 1 |
 A = 1 | X | 1 | 0 | X |

Grouping

  • Quad (m0, m1, m3, m2): the whole row A = 0, using the don't care m3, gives A'.
  • Quad (m0, m1, m4, m5): B = 0 (columns 00 and 01), using the don't care m4, gives B'.

All the 1s (0, 1, 2, 5) are covered.

Simplified: F = A' + B'

NAND-only implementation

By De Morgan's theorem: A' + B' = (A·B)'. That is a single 2-input NAND gate.

 A ---+
      [NAND]---- F = (AB)' = A' + B'
 B ---+

Check: F = 0 only for A = B = 1 (m7), which is the only required 0. So one NAND gate is enough.

✓ Answer checked by Claude Opus 5.5 (high effort)

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