# 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
    $\ f = \overline{z} \vee xy \vee \overline{y}t$.