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.

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