# How do I describe the set of all sets that are the result of a given number of changes to a given set?

Let $k$ be the number of elements that are allowed to change in a set with fixed size $n$. How would one formally describe the set of all possible sets that are a result of changing $k$ elements of the given set?

As an example, for $k = 1$, I came up with this:

$\eta(\{s_1,s_2,...,s_n\}) = \{ \{s_1,..., s_{i-1}, s', s_{i+1}, ... ,s_n\} | \forall i \in N, \forall s' \in S_i \}$

In which $N = \{1,2,...,n\}$ and $S_i$ is a finite set. A similar approach can be used for any $k$ as long as you consider it fixed. But how can I describe the set if $k$ is a variable?

The tag of this question is most likely to be irrelevant but I didn't find a more suitable one.

• What is the part after the "|" supposed to mean? Commented Apr 11, 2016 at 3:07
• @YoTengoUnLCD $\eta({s})$ is the set of sets in which each element ($\forall i \in N$) of $s$, all its possible values ($\forall s' \in S_i$) are considered. Commented Apr 11, 2016 at 9:17