Kuratowski's Closure-Complement Problem/Complement

Theorem
The complement of $A$ in $\R$ is given by:


 * 14Sets-Complement.png

Proof
For ease of analysis, let:
 * $A_1 := \left\{ {\frac 1 n : n \in \Z_{>0} }\right\}$
 * $A_2 := \left({2 \,.\,.\, 3}\right)$
 * $A_3 := \left({3 \,.\,.\, 4}\right)$
 * $A_4 := \left\{ {4 \tfrac 1 2} \right\}$
 * $A_5 := \left[{5 \,.\,.\, 6}\right]$
 * $A_6 := \left\{ {x \in \Q: 7 \le x < 8}\right\}$

Thus:
 * $\displaystyle A = \bigcup_{i \mathop = 1}^6 A_i$

By De Morgan's Laws:


 * $\displaystyle A' := \R \setminus A = \bigcap_{i \mathop = 1}^6 \left({\R \setminus A_i}\right)$

from which the result follows by inspection.