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

## Notation and basic notions

1. Define a **set**. What two properties distinguish a set from an arbitrary
   list, and why do $\{1,2,2,3\}$ and $\{3,2,1\}$ both denote the same set as
   $\{1,2,3\}$?
2. Give the **roster** form and a **set-builder** form of the set of even numbers
   between $1$ and $10$. When is set-builder form the only practical choice?
3. What is the **cardinality** $|A|$? What is $|\varnothing|$? When is
   $A \subseteq B$, and how is set **equality** proved from inclusions?

## The operations

4. State the formal definitions of $A \cup B$, $A \cap B$, $A \setminus B$, and
   $\overline{A}$.
5. Why does the **complement** $\overline{A}$ require a **universal set** $U$,
   while union, intersection, and difference do not?
6. Is $A \setminus B = B \setminus A$ in general? Give a counterexample using the
   sets $A = \{2,4,6,8,10,12\}$, $B = \{3,6,9,12\}$ of Tasks 6–9.
7. Express $A \setminus B$ through intersection and complement, and use it to
   explain why $\overline{A \setminus B} = \overline{A} \cup B$.

## Diagrams

8. For two sets $A, B$ a Venn diagram has four regions. Name each region and the
   expression it denotes.
9. Describe, step by step, how you would **shade** $A \setminus (B \cup C)$ on a
   three-circle diagram (as in Task 1).
10. What is the difference between an **Euler** diagram and a **Venn** diagram?
    When would you draw circle $A$ *inside* circle $B$?

## Algebra and proofs

11. State the two **De Morgan laws** for sets and the **distributive** law of
    $\cap$ over $\cup$.
12. Prove $A \setminus (A \setminus B) = A \cap B$ algebraically, citing one law
    per step (as in Task 4).
13. Verify the identity $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ by the
    **diagram** method, and state which single algebraic law it expresses.

## Set-builder and power set

14. List the elements of $\{\, x \mid x \in \mathbb{N},\ x \ \text{is a divisor of
    } 12 \,\}$ and of $\{\, x \mid x \in \mathbb{N},\ 10 < x < 20,\ x \
    \text{is prime} \,\}$.
15. What is the **power set** $\mathcal{P}(A)$? How many elements does it have when
    $|A| = 4$? List $\mathcal{P}(\{a, b\})$ in full.