Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 4 / 2D / b / i

The first condition concerns the domains: their symmetric difference $A△B$ must be a finite set. The original PDF prints this condition and the agreement condition together; the converted TeX omits both. The two existing solution scopes are used here to prove the domain and value parts of the single equivalence relation assertion.

Domain agreement is a reflexive relation because $A△A=\varnothing$, and a symmetric relation because $A△B=B△A$. For the transitive relation property, membership can differ between $A$ and $C$ only if it differs between $A$ and $B$ or between $B$ and $C$. ThusA set union of two finite sets is finite, so the domain condition passes through a chain of related pairs. This is the domain part of equivalence of partial functions modulo finite changes.