Exercise 1:

Let:

• \(A = \{1, 2, 3\}, \quad B = \{-2, -1, 2, 3, 4\}, \quad E = \{-3, -2, -1, 0, 1, 2, 3, 4, 5\}\).

Describe the following sets:

• \(A \cap B, \quad A \cup B, \quad A \setminus B, \quad B \setminus A, \quad A \Delta B, \quad \complement_E A, \quad \complement_E B\).

Let:

• \(A = ]-\infty, 3], \quad B = ]-2, 7]\).

Determine:

• \(A \cap B, \quad A \cup B, \quad \complement_\mathbb{R} A, \quad \complement_\mathbb{R} B\).

Exercise 2:

Let \(A\), \(B\), \(C\) be three subsets of a set \(E\). Prove the following relations:

• \(\complement_E (A \cap B) = \complement_E A \cup \complement_E B\).
• \(A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\).
• \(A \subseteq B \iff \complement_E B \subseteq \complement_E A\).
• Show that: \[ \begin{cases} A \cup B = A \cup C \\\\ A \cap B = A \cap C \end{cases} \iff B = C. \]

Exercise 3:

Given a set \(E\), show that \(\forall A, B \in \mathcal{P}(E)\), if \(A \cap B = A \cup B\), then \(A = B\).