Not every boolean function is constructed from $\wedge$ (and) and $\vee$ (or)
I'm not sure you understood what was asked in this problem. The aim is to find a function $f$ that cannot be built with the symbols $∨$ or $∧$ only.
Your first solution is the function $f: (p, q) \mapsto p∧q$ which is constructed exactly with the symbol $∧$, so it does not answer the problem. You have the same problem with the second solution you provided.
Try to find other logical symbols you could use to construct a function, and with a good choice, prove that it can not be expressed solely with $∨$ and $∧$.