Infinity And Complements

Suppose \(X \subseteq \mathbb{N}\)

Let \(A\) be the statement that \(X\) contains finitely many elements.

Let \(B\) be the statement that the complement of \(X\), say \(X'= \mathbb{N} \setminus X\) contains infinitely many elements.

Which of the following implications is/are true?

(1): \(A \implies B\).
(2): \(B \implies A\).

Notation: \(\mathbb N \) denotes the set of natural numbers.


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