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2.7: Exercises

  • Page ID
    27623
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    1. DeMorgan’s Law:
      1. Show that DeMorgan’s Law is correct for two variables, (A’B’)’ = A+B by using a truth table.
      2. Does DeMorgan’s Law hold for 3 variables? What about 4 variables?
    2. Simplify the following Boolean expressions using Boolean algebra
      1. A + AB
      2. A + A’B
      3. AB’C’ + ABC’
      4. AC + AC’ + A’B + A’B’
    3. The operations AND, OR, and NOT are universal in that any Boolean function can be implemented using just these three gates.
      1. Prove by construction that the NAND gate is universal by creating AND, OR, and NOT gates using only the NAND gate.
      2. Prove by construction that the NOR gate is also universal.
      3. Why are the AND and OR gate not universal? (e.g. what simple operation cannot be created with just an AND or OR gate?)
    4. For inputs A and B, show how to use an XOR gate to create a NOT gate if B is 1, and a buffer if B is 0.
    5. XOR is sometimes called an “odd” function because the result of an XOR is 1 if the number of 1’s the minterm is odd, the xor is 1, otherwise it is 0. Show that this is true for 3 and 4 variable XOR functions, e.g. A ⊕ B ⊕ C, and A ⊕ B ⊕ C ⊕ D.
    6. For the following truth table:
      1. Give the DNF equation for the table.
      2. Minimize the equation using a K-map.
      3. Show that the DNF is equivalent to the minimum representation using Boolean algebra.

        A

        B

        C

        F(A,B,C)

        0 0 0 0
        0 0 1 1
        0 1 0 1
        0 1 1 0
        1 0 0 0
        1 0 1 1
        1 1 0 1
        1 1 1 0
    7. For the following truth table
      1. Give the DNF equation for the table.
      2. Minimize the equation using a K-map.
      3. Show that the DNF is equivalent to the minimum representation using Boolean algebra.

        A

        B

        C

        F(A,B,C)

        0 0 0 1
        0 0 1 1
        0 1 0 1
        0 1 1 1
        1 0 0 0
        1 0 1 1
        1 1 0 0
        1 1 1 1
    8. For the following truth table
      1. Give the DNF equation for the table.
      2. Minimize the equation using a K-map.
      3. Show that the DNF is equivalent to the minimum representation using Boolean algebra.

        A

        B

        C

        F(A,B,C)

        0 0 0 1
        0 0 1 0
        0 1 0 0
        0 1 1 1
        1 0 0 1
        1 0 1 0
        1 1 0 0
        1 1 1 1
    9. Solve the 7-segment display problem for segments b, d, and f.

    This page titled 2.7: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Charles W. Kann III via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.