Is the Notion of the Empty Set Relative or Absolute? Suppose we specify subsets of a reference set by pairs, where the first co-ordinate specifies a member of the universe of discourse, and the second co-ordinate specifies the value that the characteristic function yields for that member of the reference set.  
E. G., if we have $\{a, b\}$ as the universe of discourse, then we have subsets $\{(a, 0), (b, 0)\}, \{(a, 0), (b, 1)\}, \{(a, 1), (b, 0)\}, \{(a, 1), (b, 1)\}$, where $(x, 0)$ indicates that $x$ does not belong to the set, while $(x, 1)$ indicates that $x$ belongs to the set.
Now, consider the empty set compared across different universes of discourses, for example $\{a, b\}$, and $\{a\}$.
For $\{a, b\}$ we have $\{(a, 0), (b, 0)\}$ as the empty set under this specification, and for $\{a\}$ we have $\{(a, 0)\}$ as the empty set under this specification.  
So, it would seem that we have a relative notion of an empty set in this context, but oftentimes books talk of "the empty set" which suggests it as absolute.  
So, does the concept of the empty set come as absolute, or relative?

It is absolute. What you're seeing is an image of the empty set under a map and this can be different for different maps.
