Modeling the coin weighting problem

Suppose we have n coins with weights 0 or 1 and a scale for weighting them. We would like to determine the weight of each coin by minimizing the number of weightings.

The book that I am reading states that the above problem can be modeled in the following way. We say that the subsets ${S}_{1},\dots ,{S}_{m}$ of {1,…,n} are determing if any $T\subseteq \{1,\dots ,n\}$ can be uniquely determined by the cardinalities $|{S}_{i}\cap T|$ for $1\le i\le m.$. The minimum number of weightings is then equivalent to the least m for which a determing sequence of sets exists.

My question is. How exactly does this reduce to the coin weighting problem?

Suppose we have n coins with weights 0 or 1 and a scale for weighting them. We would like to determine the weight of each coin by minimizing the number of weightings.

The book that I am reading states that the above problem can be modeled in the following way. We say that the subsets ${S}_{1},\dots ,{S}_{m}$ of {1,…,n} are determing if any $T\subseteq \{1,\dots ,n\}$ can be uniquely determined by the cardinalities $|{S}_{i}\cap T|$ for $1\le i\le m.$. The minimum number of weightings is then equivalent to the least m for which a determing sequence of sets exists.

My question is. How exactly does this reduce to the coin weighting problem?