$\lambda A\in \wp(\mathbb{N}).\mathbb{N} \setminus A$
I have the following lambda function, and I want to determine its image - clearly it maps to $\wp(\mathbb{N})$ but the image must also contain all singletons of a singeltons - for example $\{\{1\}\}$, $\{\{2\}\}$ etc..
Is it correct ? and how do I define this image mathematically?
If we apply the function to $A$, where $A$ is an element of $\wp(\mathbb{N})$, that is, a set of natural numbers, it will return the set complement of $A$ wrt. $\mathbb{N}$. This, too, must be a set of natural numbers. Moreover, the function is easily seen to be surjective. So $\text{Im}\;(\lambda A. \mathbb{N} \setminus A) = \wp(\mathbb{N})$.