Determine the number of proper subsets contained in s if s = {x|x is an even, negative integer and x > -27}

First, lets determine  (the number of elements in ).

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 .

A proper subset of  is can't be  itself, this means the a proper subset of  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  with 1 element are given by  (which means 13 combine 1), in the same way the number of subsets with 2 elements is given by  and so forth until reaching the number of subsets with 12 elements.

The previous statement can be written as:


In the previous summation we also included the empty subset, which is always a proper subset of .

So is an acceptable answer, but we can simplify it. 
The following statement is true (it can be easily proven):


From our previous discussion, this means that the number of subsets of any arbitrary set  is equal to .
This means that the number of proper subsets (which is the number of subset minus the set itself, for finite subsets) is equal to .

So, the answer is:


There are 8191 proper subsets of .