Exercise 1.2.3.
Decide which of the following represent true statements about the nature of sets. For any that are false,
provide a specific
example where statement in question does not hold.
- If \(A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \cdots \) are all sets containing
an infinite number of elements, then the intersection \( \bigcap_{n=1}^\infty A_n \) is infinite as
well.
- If \(A_1 \supseteq A_2 \supseteq A_3 \supseteq A_4 \cdots \) are all finite, nonempty
sets of real numbers, then the intersection \( \bigcap_{n=1}^\infty A_n \) is finite and nonempty.
- \(A \cap (B \cup C) = (A \cap B) \cup C \)
- \(A \cap (B \cap C) = (A \cap B) \cap C \)
- \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
Solution:
(a). This is false. Let \(A_i = \{x \in \mathbb{N} : x > i\}\). Suppose that, for contradiction, there
exists \(x \in \bigcup_{n=1}^\infty A_n \). This implies that \(x > i\) for all \(i \in \mathbb{N}\).