Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 16F / b / iii / Solution

Statement (iii) can fail because a nontransitive ambient set can omit the elements that distinguish two of its members. TakeNeither element of $y$ has an element that also lies in $y$: the second contains $\{\varnothing \}$, which is omitted. Their predecessor sets for membership restricted to $y$ are therefore both empty, although the two elements are distinct. Hence this restricted membership relation, and any isomorphic relation, is not extensional.

Solved by gpt-5.6-sol high.