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Exercises 3.3 Exercises
Creating truth tables.
In each of
Exercises 1–2, write out the truth table for the given boolean polynomial.
1.
\(p(x,y) = (x \lgcand y)' \lgcand x' \text{.}\)
2.
\(q(x,y,z) = (x \lgcor y)' \lgcand (z \lgcor x) \lgcand y \text{.}\)
3.
Explain why the boolean polynomial
\(p(x,y) = x \lgcor y \lgcor y' \) is
not in disjunctive form.
Disjunctive normal form from a truth table.
In each of
Exercises 4–6, write out a boolean polynomial in disjunctive normal form that has the given truth table.
4.
| \(x \) |
\(y \) |
\(p(x,y) \) |
| \(1 \) |
\(1 \) |
\(1 \) |
| \(1 \) |
\(0 \) |
\(1 \) |
| \(0 \) |
\(1 \) |
\(1 \) |
| \(0 \) |
\(0 \) |
\(0 \) |
5.
| \(x \) |
\(y \) |
\(p(x,y) \) |
| \(1 \) |
\(1 \) |
\(1 \) |
| \(1 \) |
\(0 \) |
\(0 \) |
| \(0 \) |
\(1 \) |
\(1 \) |
| \(0 \) |
\(0 \) |
\(0 \) |
6.
| \(x \) |
\(y \) |
\(z \) |
\(p(x,y,z) \) |
| \(1 \) |
\(1 \) |
\(1 \) |
\(1 \) |
| \(1 \) |
\(1 \) |
\(0 \) |
\(0 \) |
| \(1 \) |
\(0 \) |
\(1 \) |
\(0 \) |
| \(1 \) |
\(0 \) |
\(0 \) |
\(0 \) |
| \(0 \) |
\(1 \) |
\(1 \) |
\(1 \) |
| \(0 \) |
\(1 \) |
\(0 \) |
\(0 \) |
| \(0 \) |
\(0 \) |
\(1 \) |
\(0 \) |
| \(0 \) |
\(0 \) |
\(0 \) |
\(0 \) |
Disjunctive normal form from a boolean polynomial.
In each of
Exercises 7–9, write out a boolean polynomial in disjunctive normal form that is equivalent to the given boolean polynomial.
7.
\(p(x,y,z) = (x \lgcor y) \lgcand z \text{.}\)
8.
\(q(x,y,z) = \bigl[(x \lgcand y') \lgcor (x \lgcand z)\bigr]' \lgcor x' \text{.}\)
9.
\(r(x,y,z) = (x \lgcand y') \lgcor (x \lgcand z) \lgcor (x \lgcand y) \text{.}\)
