relational algebra to get unique pairs
You need more carefully defined $\theta$-join and then set difference:

$$R1 := \rho_{name1/name,hobby1/hobby}(Person)$$
$$R2 := Person \bowtie_{(hobby=hobby1) \land (name \neq name1)} R1$$
$$R3 := \pi_{hobby}(Person) \setminus \pi_{hobby}(R2)$$