Is this definition of a universal set correct?

Question

Jay Cummings (2021, Proofs: A Long-Form Mathematics Textbook, p. 83):

So if for example $A=\{1,2\}$, $U_1=\{1,2,3\}$, and $U_2=\{1,2,3,4\}$, then $U_1$ and $U_2$ are both universal sets of $A$? So what's the complement of $A$?

Answer 1

The book says that the complement of $A$ in $U$ is $U \setminus A$.

So, the complement of $\{1,2\}$ in $\{1,2,3\}$ is $\{3\}$, while the complement of $\{1,2\}$ in $\{1,2,3,4\}$ is $\{3,4\}$.