Sum of Numbers of Array
Mathematica

This is easy in Mathematica. (1) Generate the subsets of your input list, using the Subsets[] command (I had originally used Permutations[], which works but produces redundant results), (2) sum the numbers in each subset by mapping the Total[] command across the list of subsets, then (3) check to see if the target sum S is represented among those sums with MemberQ[]. To be concise, I've shown this below implemented as a single nested command.

Put your inputs in the first line and execute this:

list = {0, 4, 3, 8, 1}; target = 6;
MemberQ[Total /@ Subsets[list], target]


Assuming one-character variable names, that's 28 bytes of code.