Consider any one segment of a 7-segment LED display?

Let's say I use an and gate as an input. And someone else uses a not gate and gate and not gate again? Are they logically the same? And for nor or nand gates yes they can create any other gate. If I take a nor gate and tie its inputs together I create an not gate. Are they the same no but functionally identical and logically identical as well

Say that two different people each design their own version of a circuit to control that segment. They agree on which input bit patterns (patterns of T and F) represent which of the inputs 0-9. They also agree on when the LED should turn on given each of those inputs. However, the circuits they produce use very different gates. Could two such different circuits both be correct? If so, can the circuits be logically equivalent? If so, must the circuits be logically equivalent?