The ID3 algorithm constructs a decision tree from the data based on the information gain. In the beginning, we start with the set S. The data items in the set S have various properties according to which we can partition the set S. If an attribute A has the values {v1, ..., vn}, then we partition the set S into the sets S1, ..., Sn, where the set Si is a subset of the set S, where the elements have the value vi for the attribute A.
If each element in the set S has attributes A1, ..., Am, then we can partition the set S according to any of the possible attributes. The ID3 algorithm partitions the set S according to the attribute that yields the highest information gain. Now suppose that it was attribute A1. Then for the set S we have the partitions S1, ..., Sn, where A1 has the possible values {v1,..., vn}.
Since we have not constructed...