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Minimize 9-term SOP and implement in NAND

Short answer

The minimized expression is F = A'B + AD' + AB'C'. Implement with NAND gates: use NAND inverters for A', B', C', D', then three NAND gates for (A' B)', (A D')', and (A B' C')', and a final NAND to combine them.

Minimize the expression and implement the reduced expression by using NAND gates. F = A'BC'D' + A'BC'D + A'BCD + A'BCD' + AB'C'D' + AB'C'D + AB'CD' + ABC'D' + ABCD'

Minterm numbers

Write each term as a binary number ABCD:

TermABCDMinterm
A‾BC‾ D‾\overline{A}B\overline{C}\,\overline{D}01004
A‾BC‾D\overline{A}B\overline{C}D01015
A‾BCD\overline{A}BCD01117
A‾BCD‾\overline{A}BC\overline{D}01106
AB‾ C‾ D‾A\overline{B}\,\overline{C}\,\overline{D}10008
AB‾ C‾DA\overline{B}\,\overline{C}D10019
AB‾CD‾A\overline{B}C\overline{D}101010
ABC‾ D‾AB\overline{C}\,\overline{D}110012
ABCD‾ABC\overline{D}111014
F=Σm(4,5,6,7,8,9,10,12,14)F = \Sigma m(4,5,6,7,8,9,10,12,14)

K-map minimization

        CD
AB      00  01  11  10
      +---+---+---+---+
 00   | 0 | 0 | 0 | 0 |
      +---+---+---+---+
 01   | 1 | 1 | 1 | 1 |
      +---+---+---+---+
 11   | 1 | 0 | 0 | 1 |
      +---+---+---+---+
 10   | 1 | 1 | 0 | 1 |
      +---+---+---+---+
  • Quad m(4,5,6,7) → A‾B\overline{A}B
  • Quad m(8,10,12,14) → AD‾A\overline{D}
  • Pair m(8,9) → AB‾ C‾A\overline{B}\,\overline{C}
F=A‾B+AD‾+AB‾ C‾F = \overline{A}B + A\overline{D} + A\overline{B}\,\overline{C}

Check: row 01 is covered by A‾B\overline{A}B; 8, 10, 12, 14 by AD‾A\overline{D}; 9 (with 8) by AB‾ C‾A\overline{B}\,\overline{C}. The 9-term, 36-literal expression reduces to 3 terms and 7 literals.

NAND implementation

An SOP expression maps directly to a two-level NAND-NAND circuit:

F=A‾B‾⋅AD‾‾⋅AB‾ C‾‾‾F = \overline{\overline{\overline{A}B}\cdot\overline{A\overline{D}}\cdot\overline{A\overline{B}\,\overline{C}}} 
A ──[NAND]── A'   (inputs tied)
B ──[NAND]── B'   (inputs tied)
C ──[NAND]── C'   (inputs tied)
D ──[NAND]── D'   (inputs tied)

A', B     ──[NAND]── P1  ─┐
A, D'     ──[NAND]── P2  ─┤──[NAND]── F
A, B', C' ──[NAND]── P3  ─┘

Gates: 4 NAND inverters (for A‾,B‾,C‾,D‾\overline{A}, \overline{B}, \overline{C}, \overline{D}), three first-level NANDs (two 2-input, one 3-input) and one 3-input output NAND.

✓ Answer checked by Claude Opus 5.5 (high effort)

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