Question

$$\bigcap\{\mathbb{R}\setminus (a,b)|(\mathbb{N}\subseteq \mathbb{R}\setminus (a,b))(a,b\in\mathbb{R})\}=\mathbb{N}$$ olduğunu gösteriniz.

Answer 1

$\mathcal{A}=\{\mathbb{R}\setminus (a,b)|(\mathbb{N}\subseteq\mathbb{R}\setminus (a,b))(a,b\in\mathbb{R})\}$ diyelim.
$$\mathbb{N}\subseteq \bigcap\mathcal{A}\ldots (1)$$ olduğu aşikar. $$\bigcap\mathcal{A}\subseteq\mathbb{N}\ldots (2)$$ olduğunu gösterelim.
$$\{\mathbb{R}\setminus (-n,0)|n\in\mathbb{N}\}\cup \{\mathbb{R}\setminus (n,n+1)|n\in\mathbb{N}\}\subseteq \mathcal{A}$$ $$\Rightarrow$$ $$\bigcap \mathcal{A}\subseteq \bigcap\left(\{\mathbb{R}\setminus (-n,0)|n\in\mathbb{N}\}\cup \{\mathbb{R}\setminus (n,n+1)|n\in\mathbb{N}\}\right)=\mathbb{N}\ldots (2)$$

$$(1),(2)\Rightarrow \bigcap\mathcal{A}=\mathbb{N}.$$