Question 107
Let $A = \{ a = q + \frac{1}{q} \mid q \in \mathbb{Q}^*, q \gt 0 \}$,
$$A + A = \{a + b \mid a, b \in A\}, \quad A \cdot A = \{a \cdot b \mid a, b \in A\}.$$
Prove that:
i) $A + A \neq A \cdot A$;
ii) $(A + A) \cap \mathbb{N} = (A \cdot A) \cap \mathbb{N}$.
$$A + A = \{a + b \mid a, b \in A\}, \quad A \cdot A = \{a \cdot b \mid a, b \in A\}.$$
Prove that:
i) $A + A \neq A \cdot A$;
ii) $(A + A) \cap \mathbb{N} = (A \cdot A) \cap \mathbb{N}$.