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Challenge Problems: Quine-McCluskey Method

These challenge problems test deeper understanding. Only final answers are provided — work through each problem on your own.

Challenge 1: QM Method with Don't Cares

Use the Quine-McCluskey method to find the minimum SOP expression for:

\[F(A, B, C, D) = \sum m(0, 1, 2, 5, 6, 7, 8, 9, 14) + \sum d(3, 11, 15)\]
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Minterms + don't cares: {0, 1, 2, 3, 5, 6, 7, 8, 9, 11, 14, 15}

Group 0 (0 ones): 0000

Group 1 (1 one): 0001, 0010, 1000

Group 2 (2 ones): 0011, 0101, 0110, 1001

Group 3 (3 ones): 0111, 1011, 1110

Group 4 (4 ones): 1111

Minimum SOP: \(F = \overline{B}\,\overline{C} + \overline{A}\,D + \overline{A}\,B\,C + BCD\)

Challenge 2: Find All Prime Implicants and Essential PIs

For \(F(A, B, C, D) = \sum m(0, 4, 5, 6, 7, 8, 9, 14, 15)\), use the Quine-McCluskey method to:

  1. List all prime implicants
  2. Identify the essential prime implicants
  3. Determine the minimum SOP
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Step 1 — All prime implicants:

PI Minterms Covered Expression
PI1 4, 5, 6, 7 \(\overline{A}\,B\)
PI2 14, 15 \(A\,B\,C\)
PI3 8, 9 \(A\,\overline{B}\,\overline{C}\)
PI4 0, 8 \(\overline{B}\,\overline{C}\,\overline{D}\)

Prime implicants: \(\overline{A}\,B\); \(A\,B\,C\); \(A\,\overline{B}\,\overline{C}\); \(\overline{B}\,\overline{C}\,\overline{D}\)

Step 2 — Essential PIs:

  • \(\overline{A}\,B\) is essential (only PI covering 5, 6, 7)
  • \(A\,B\,C\) is essential (only PI covering 14, 15)
  • \(A\,\overline{B}\,\overline{C}\) is essential (only PI covering 9)

After selecting essentials: minterms 4, 5, 6, 7, 8, 9, 14, 15 are covered. Minterm 0 remains.

\(\overline{B}\,\overline{C}\,\overline{D}\) covers minterm 0.

Step 3 — Minimum SOP: \(F = \overline{A}\,B + A\,B\,C + A\,\overline{B}\,\overline{C} + \overline{B}\,\overline{C}\,\overline{D}\)

Challenge 3: PI Chart with Cyclic Cover Problem

For \(F(W, X, Y, Z) = \sum m(0, 1, 5, 7, 8, 10, 14, 15)\), find all prime implicants using QM, construct the PI chart, and identify any cyclic (non-essential) cover situation. Find the minimum cover.

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Prime implicants:

  • \(\overline{X}\,\overline{Y}\,\overline{Z}\) (covers 0, 8)
  • \(\overline{W}\,\overline{X}\,\overline{Y}\) (covers 0, 1)
  • \(\overline{W}\,X\,Z\) (covers 5, 7)
  • \(W\,\overline{X}\,\overline{Z}\) (covers 8, 10)
  • \(W\,X\,Y\) (covers 14, 15)
  • \(X\,Y\,\overline{Z}\) (covers 10, 14)
  • \(\overline{W}\,\overline{Y}\,Z\) (covers 1, 5)

After constructing the PI chart, essential PIs are: \(W\,X\,Y\) (only cover for 15), \(\overline{W}\,X\,Z\) (only cover for 7).

Minimum SOP: \(F = \overline{X}\,\overline{Y}\,\overline{Z} + \overline{W}\,X\,Z + W\,X\,Y + W\,\overline{X}\,\overline{Z}\)

Alternatively: \(F = \overline{W}\,\overline{X}\,\overline{Y} + \overline{W}\,X\,Z + X\,Y\,\overline{Z} + W\,\overline{X}\,\overline{Z} + W\,X\,Y\,Z\)

Minimum cover (4 terms): \(F = \overline{X}\,\overline{Y}\,\overline{Z} + \overline{W}\,X\,Z + W\,\overline{X}\,\overline{Z} + W\,X\,Y\)

Challenge 4: Compare QM Result with K-Map

For \(F(A, B, C, D) = \sum m(1, 3, 4, 5, 9, 11, 12, 14)\), solve using both the Quine-McCluskey method and a K-map. Verify that both approaches yield the same minimum SOP expression.

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By K-map and QM (both methods):

Prime implicants (by QM and K-map):

  • \(\overline{B}\,D\) (covers 1, 3, 9, 11)
  • \(\overline{A}\,B\,\overline{C}\) (covers 4, 5)
  • \(B\,\overline{C}\,\overline{D}\) (covers 4, 12)
  • \(A\,B\,C\,\overline{D}\) (covers 14)

Minimum SOP: \(F = \overline{B}\,D + \overline{A}\,B\,\overline{C} + B\,\overline{C}\,\overline{D} + A\,B\,C\,\overline{D}\)

Coverage: \(\overline{B}\,D\) covers {1,3,9,11}; \(\overline{A}\,B\,\overline{C}\) covers {4,5}; \(B\,\overline{C}\,\overline{D}\) covers {12}; \(A\,B\,C\,\overline{D}\) covers {14}.

Both K-map and QM confirm this result.

Challenge 5: Petrick's Method Application

For \(F(A, B, C, D) = \sum m(2, 3, 7, 9, 11, 13)\), after finding all prime implicants via QM, use Petrick's method to find all minimum SOP covers.

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Prime implicants:

  • \(P_1\): \(\overline{A}\,\overline{B}\,C\) (covers 2, 3)
  • \(P_2\): \(\overline{A}\,C\,D\) (covers 3, 7)
  • \(P_3\): \(\overline{B}\,C\,D\) (covers 3, 11)
  • \(P_4\): \(A\,\overline{B}\,D\) (covers 9, 11)
  • \(P_5\): \(A\,\overline{C}\,D\) (covers 9, 13)

Petrick's method: Cover each minterm:

  • \(m_2\): \(P_1\)
  • \(m_7\): \(P_2\)
  • \(m_{13}\): \(P_5\)

Essential PIs: \(P_1\), \(P_2\), \(P_5\) (each is the only cover for a minterm).

After essentials: \(P_1\) covers {2,3}, \(P_2\) covers {3,7}, \(P_5\) covers {9,13}. Remaining: \(m_{11}\).

\(m_{11}\) covered by \(P_3\) or \(P_4\).

Two minimum covers:

  1. \(F = \overline{A}\,\overline{B}\,C + \overline{A}\,C\,D + A\,\overline{C}\,D + \overline{B}\,C\,D\)
  2. \(F = \overline{A}\,\overline{B}\,C + \overline{A}\,C\,D + A\,\overline{C}\,D + A\,\overline{B}\,D\)