Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 16J / c / Solution

Taking transitive closure adds only descendants of existing members, whose ranks are lower. Thus the supremum of member ranks, and hence the rank of the set, is unchanged: $\rho (x)=\rho (\mathrm{tcl}x)$.

Solved by gpt-5.6-sol high.