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 .
