Properties of the operations with intervals

Some important properties of the operations with intervals are as follows; given $J$ and $K$ two intervals any:

• $J \subseteq (J\cup K)$ and $K \subseteq (J\cup K)$
• $(J\cap K) \subseteq J$ and $(J\cap K) \subseteq K$
• if $J \subseteq K$ then $\overline{K} \subseteq \overline{J}$
• $\overline{J\cup K}=\overline{J}\cap\overline{K}$ and $\overline{J\cap K}=\overline{J}\cup\overline{K} $
• $\overline{(\overline{K})}=K$, and in particular, as $\overline{\emptyset}=\mathbb{R},$ we have $\overline{\mathbb{R}}=\overline{\overline{\emptyset}}=\emptyset.$

In addition, regarding the length of the intervals we have:

• if $J\subseteq K$ then $long(J)\leq long(K)$
• $long(J\cup K) \leq long(J) + long(K)$
• $max(long(J),long(K)) \leq long(J\cup K)$
• $long(J\cap K) \leq max(long(J),long(K))$
• $long(J)$ is finite iff $long(\overline{J})$ is not.