= Solution
For $q=1,\ldots,5$, choose the first $q$ elements of $\{1,2,4,8,16\}$. Every <sum of a subset> has a unique <binary representation>, so distinct <subsets> cannot have the same sum. To see this directly, take the largest power of two belonging to exactly one of two proposed <subsets>. It exceeds the sum of all smaller powers, and therefore cannot be cancelled by their difference. Thus these chosen sets are <dissociated sets> and do not have the two-sum property.
If cardinality zero is included, the empty <set> has only one <subset>, so it cannot supply two distinct ones either. \b[There is a <counterexample> at every cardinality from $0$ through $5$.]
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