Can a set containing $0$ be purely imaginary? A purely imaginary number is one which contains no non-zero real component. If I had a sequence of numbers, say $\\{0+20i, 0-i, 0+0i\\}$, could I call this purely imaginary? My issue here is that because $0+0i$ belongs to multiple sets, not just purely imaginary, is there not a valid case to say that the sequence isn't purely imaginary?

0 is both purely real and purely imaginary. The given set is purely imaginary. That's not a contradiction since "purely real" and "purely imaginary" are not fully incompatible. Somewhat similarly baffling is that "all members of X are even integers" and "all members of X are odd integers" is not a contradiction. It just means that X is an empty set.