Axiomatic structure/Abstract and examples/Remark

Abstract structures like a set, a mapping, a binary operation, or a group have a double life. On one hand, they are really just the given formal structure; the elements are just some elements in a somehow given set, a binary operation is just any binary operation, and one should not imagine anything concrete. The symbols chosen are arbitrary and without any meaning. On the other hand, these abstract structures gain a second life in that many concrete mathematical structures obey the abstract properties. These concrete structures are examples or models for the abstract structure (and they are also a motivation to introduce the abstract structure). Both viewpoints are important, and one should always try not to confuse them.