Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$

Question

Find the inverse of $θ:P(\Bbb{Z})→P(\Bbb{Z})$ defined as $θ(X) = \bar X$ (the complement of $X$)?

Would the inverse of the function just be the function itself?

Answer 1

The inverse function $\theta^{-1}$ is just the function $\theta$ itself, since the complement of the complement of a set $X$ is the set $X$ itself. For example, the complement of $X=$ the set of even integers is the set of odd integers. The complement of the latter set $\bar X$, i.e., the set of integers which are not odd, is the set of even integers $=X$.