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You have a function best(int n,set of intervals). If the given set of intervals covers the 0th through n-1th numbers, then best(n,set of intervals) is the minimum, over all intervals (call each one I in turn) containing the nth number, of best(n+1,set of intervals \[Union] I).

best(int n,set of ...

I have not done the minimal set problem, and have only thought about it for two minutes, but here is how I would do it

Have a function
best(next number to consider,chosen set of intervals)
= minimum over all intervals I that cover the next number to consider, of best(next number to consider+1 ...

I am also facing the same problem. It is difficult to formulate the solutions with DP sometimes. I read some examples in introduction to algorithms , it seems I had mastered the idea of DP, actually find out not when trying to solve other problems myself. Normally it is said as that book "beneath ...