Let $L$ and $E$ be relations defined as
$L = \{ (s,r) \in S \times R \mid \text{student } s \text{ lives in dorm room } r \}$
$E= \{(s,c) \in S \times C \mid \text{student } s \text{ is enrolled in course } c\}$
where $S$ is the set of students in a school, $C$ is the set of courses in a school and $R$ is the set of all dorm rooms.
I have to find out what $L^{-1} \circ L$ and $E \circ (L^{-1} \circ L)$ mean. Now after writing out definitions I write the answer as
$L^{-1} \circ L = \{(s_1,s_2) \mid \text{there is a dorm room which is shared by two students} \}$
$ E \circ (L^{-1} \circ L) = \{(s,c) \mid \text{there is a student who has taken a course} \\ \text{ and lives with some other student in some room} \}$
Please check if it is correct. Thanks