I was reading Frank Harary's text and comparing it with Wikipedia's definition. Harary defined the family set $F$ whose set elements are vertices of the intersection graph to be distinct while Wikipedia defined $F$ to allow non-distinct set elements. What am I not understanding? Sorry if this is a silly question.
Source: https://en.m.wikipedia.org/wiki/Intersection_number_(graph_theory)
Definitions are not set in stone. Whoever wrote the Wikipedia article must think that the definition is better if we allow repeated subsets, while Harary does not. Just keep this fact in mind while reading any literature where the author talks about intersection graphs that there is a difference of opinion as to whether duplicate subsets should be allowed in the vertex set. Usually, though, the precises definition will not matter that much, or will be clear from the context.