Past exam of the year 1 of the computer science course of the University of Oxford / 2019 / Discrete mathematics, continuous mathematics, and probability / 1 / b

The set difference $\varnothing \setminus \varnothing$ is the empty set. Its power set contains exactly one element, the empty set itself. Therefore the cardinality is $1$, not zero.

A binary relation on $A$ is any subset of the Cartesian product $A\times A$. There are $n^2$ ordered pairs, and each can independently be included or omitted. Hence the number of binary relations is $2^{n^2}$. This also gives one relation when $n=0$, namely the empty set.