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Chapter 12: Boolean Algebra

Expert-verified
Discrete Mathematics and its Applications
Pages: 811 - 844
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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151 Questions for Chapter 12: Boolean Algebra

  1. Show that a Boolean function can be represented as a Boolean product of maxterms. This representation is called the product-of-sums expansion or conjunctive normal form of the function. (Hint: Include one maxterm in this product for each combination of the variables where the function has the value 0.)

    Found on Page 822

  2. Draw the \({\bf{3}}\)-cube \({{\bf{Q}}_{\bf{3}}}\) and label each vertex with the minterm in the Boolean variables \({\bf{x, y}}\), and \({\bf{z}}\) associated with the bit string represented by this vertex. For each literal in these variables indicate the \({\bf{2}}\)-cube \({{\bf{Q}}_{\bf{2}}}\) that is a subgraph of \({{\bf{Q}}_{\bf{3}}}\) and represents this literal.

    Found on Page 842

  3. Construct a circuit for a half subtractor using AND gates, OR gates, and inverters. A half subtractor has two bits as input and produces as output a difference bit and a borrow.

    Found on Page 828

  4. How many different Boolean functions are there of degree \(7\)\({\bf{?}}\)

    Found on Page 818

  5. \(a)\)Explain how \(K{\bf{ - }}\)maps can be used to simplify sum-of-products expansions in three Boolean variables.

    Found on Page 844

  6. Draw the Hasse diagram for the poset consisting of the set of the \({\bf{16}}\)Boolean functions of degree two (shown in Table \({\bf{3}}\) of Section \({\bf{12}}{\bf{.1}}\)) with the partial ordering \( \le \).

    Found on Page 844

  7. Construct a circuit for a full subtractor using AND gates, OR gates, and inverters. A full subtractor has two bits and a borrow as input, and produces as output a difference bit and a borrow.

    Found on Page 828

  8. Find the product-of-sums expansion of each of the Boolean functions in Exercise 3.

    Found on Page 822

  9. Prove the absorption law \({\bf{x + xy = x}}\) using the other laws in Table \(5\).

    Found on Page 818

  10. \({\bf{a)}}\)Explain how \({\bf{K}}\)-maps can be used to simplify sum-of products expansions in four Boolean variables.

    Found on Page 844

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