6. Control questions and tasks / Контрольні запитання і завдання #

Canonical forms #

1. Define a minterm and a maxterm of a row. State the polarity rule — when is a variable complemented in each?
2. How do you read the canonical DNF and the canonical CNF off a truth table? Which rows does each use?
3. Write the canonical DNF and the canonical CNF of the function that is $1$ only on the rows $(1,1,0)$ and $(0,0,1)$ (variables $x,y,z$).
4. Why are canonical forms unique but usually not minimal?

The laws and algebraic minimization #

5. State the combining law and the absorption law. Which one removes a variable, and which removes a whole term?
6. Give the three steps of algebraic minimization (eliminate $\to$; push $\neg$ inward; simplify). What is the method’s main weakness?
7. Simplify $\ xy \vee x\overline{y} \vee \overline{x}y\$ using the laws to a minimal DNF.

Karnaugh maps #

8. On a three-variable map, which cells are adjacent? Explain the wrap-around (cylinder) adjacency and why the first and last columns group.
9. A group of $2^k$ cells eliminates how many variables? How do you read the term of a group?
10. State the rule for choosing groups to obtain a minimal DNF. How do you instead obtain a minimal CNF from the map?
11. Minimize $\ f = xyz \vee \overline{x}\,\overline{y}\,\overline{z} \vee \overline{x}yz \vee xy\overline{z} \vee \overline{x}y\overline{z}\$ with a Karnaugh map (this is Задача 1 — confirm $f = y \vee \overline{x}\, \overline{z}$).
12. On a four-variable map (rows $xy$, columns $zt$, both in Gray order), how many cells can a single group contain, and which four cells does the four-corners group combine? Minimize Variant 4 of Задача 4 and confirm $\ f = \overline{z} \vee xy \vee \overline{y}t$.

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