Why can't the Klein-4 group be generated by a single element?

Question

This is stated in my notes, but with no proof to back it up and I can't seem to figure it out. Could someone please explain?

Answer 1

In the Klein-4 group, each element squares to the identity element. Thus the orders of the elements are 1 (for the identity) or 2 (for everything else). So nothing generates all 4 elements.

Answer 2

Because it can't? The identity generates only one element, and the other three elements generate only themselves and the identity. So, no element generates all four. You simply check it.