First, lets determine [tex]|S|[/tex] (the number of elements in [tex]S[/tex]).
We can see that the negative even integers that are between -27 and 0 are: -24, -22, -20, -18, -16, -14, -12, -10, -8, -6, -4, and -2.
This means [tex]|S|=13[/tex].
A proper subset of [tex]S[/tex] is can't be [tex]S[/tex] itself, this means the a proper subset of [tex]S[/tex] can't have 13 elements.
Given that re-arranging the elements of any set doesn't yield a different set, we use combinations (binomial coefficients) to get the number of subsets in a particular set.
The number of subsets of [tex]S[/tex] with 1 element are given by [tex](^{13}_{1})[/tex] (which means 13 combine 1), in the same way the number of subsets with 2 elements is given by [tex](^{13}_{2})[/tex] and so forth until reaching the number of subsets with 12 elements.
The previous statement can be written as:
[tex] \Sigma^{12}_{k=0}(^{13}_{k})[/tex]
In the previous summation we also included the empty subset, which is always a proper subset of [tex]S[/tex].
So [tex] \Sigma^{12}_{k=0}(^{13}_{k})[/tex] is an acceptable answer, but we can simplify it.
The following statement is true (it can be easily proven):
[tex]\Sigma^{n}_{k=0}(^{n}_{k})=2^n[/tex]
From our previous discussion, this means that the number of subsets of any arbitrary set [tex]S[/tex] is equal to [tex]2^{|S|}[/tex].
This means that the number of proper subsets (which is the number of subset minus the set itself, for finite subsets) is equal to [tex]2^{|S|}-1[/tex].
So, the answer is:
[tex] \Sigma^{12}_{k=0}(^{13}_{k})=2^{13}-1=8192-1=8191[/tex]
There are 8191 proper subsets of [tex]S[/tex].
