Lesson 04: Simplification of Boolean Expressions

Questions

  1. When a Karnaugh map has four rows or columns, they are numbered 00, 01, 11, 10 instead of 00, 01, 10, 11. Why?
  2. Simplify the following Boolean functions using three-variable K-maps:
    1. \(F(x,y,z) = x\,y + \overline x \,\overline y \,\overline z + \overline x \,y\,\overline z \)
    2. \(F(x,y,z) = \overline x \,\overline y + y\,z + \overline x \,y\,\overline z \)
    3. \(F(x,y,z) = \overline x \,y + y\,\overline z + \overline y \,\overline z \)
  3. Simplify the following Boolean functions:
    1. F(x, y, z) = Σ m(0, 1, 5, 7)
    2. F(x, y, z) = Σ m(1, 2, 3, 6, 7)
    3. F(w, x, y, z) = Σ m(1, 3, 7, 11, 15) + d(0, 2, 5)
    4. F(w, x, y, z) = Σ m(4, 5, 6, 7, 12) + d(0, 8, 13)
    5. F(w, x, y, z) = Σ m(1, 6, 10, 11, 12, 13, 15) + d(4, 5, 7, 8, 14)
  4. Simplify the following Boolean functions:
    1. F(A, B, C, D) = ∏ M(1, 3, 5, 7, 13, 15)
    2. F(A, B, C, D) = ∏ M(1, 3, 6, 9, 11, 12, 14)
  5. Simplify the following expression to (1) SoP and (2) PoS:
    1. \(\overline A \,\overline C + \overline B \,\overline C + B\,\overline C + A\,B\)
    2. \((\overline A + B + \overline D ) \cdot (\overline A + \overline B + \overline C ) \cdot (\overline A + \overline B + C) \cdot (\overline B + C + \overline D )\)
  6. Implement the following Boolean expression with XOR and AND gates:
    \[F = A\,\overline B \,C\,\overline D + \overline A \,B\,C\,\overline D + A\,\overline B \,\overline C \,D + \overline A \,B\,\overline C \,D\]

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