3.3: Exercises
Creating truth tables.
In each of Exercises 1–2, write out the truth table for the given boolean polynomial.
\(p(x,y) = (x \land y)' \land x' \text{.}\)
\(q(x,y,z) = (x \lor y)' \land (z \lor x) \land y \text{.}\)
Explain why the boolean polynomial \(p(x,y) = x \lor y \lor y'\) 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.
| \(x\) | \(y\) | \(p(x,y)\) |
| \(1\) | \(1\) | \(1\) |
| \(1\) | \(0\) | \(1\) |
| \(0\) | \(1\) | \(1\) |
| \(0\) | \(0\) | \(0\) |
| \(x\) | \(y\) | \(p(x,y)\) |
| \(1\) | \(1\) | \(1\) |
| \(1\) | \(0\) | \(0\) |
| \(0\) | \(1\) | \(1\) |
| \(0\) | \(0\) | \(0\) |
| \(x\) | \(y\) | \(z\) | \(p(x,y,z)\) |
| \(1\) | \(1\) | \(1\) | \(1\) |
| \(1\) | \(1\) | \(0\) | \(0\) |
| \(1\) | \(0\) | \(1\) | \(0\) |
| \(1\) | \(0\) | \(0\) | \(0\) |
| \(0\) | \(1\) | \(1\) | \(1\) |
| \(0\) | \(1\) | \(0\) | \(0\) |
| \(0\) | \(0\) | \(1\) | \(0\) |
| \(0\) | \(0\) | \(0\) | \(0\) |
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.
\(p(x,y,z) = (x \lor y) \land z \text{.}\)
\(q(x,y,z) = [(x \land y') \lor (x \land z)]' \lor x' \text{.}\)
\(r(x,y,z) = (x \land y') \lor (x \land z) \lor (x \land y) \text{.}\)