Answer
Statement "The set {1,2,3,....,1000} has 2^{1000} proper subsets" is False.
Work Step by Step
{1,2,3,....,1000}
A set of n elements have 2^{n} number of subsets.
Here set has 1000 elements. So total number of subsets are 2^{1000}
As these 2^{1000} subsets also include the one subset which is the set itself. We need to exclude this one subset to get the total number of proper subsets.
Thus Total number of proper subsets are = 2^{1000} - 1.
So the Statement "The set {1,2,3,....,1000} has 2^{1000} proper subsets" is False.
