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.