Define what is meant by
- a partially ordered set,
- a totally ordered set,
6 a well-ordered set.
(a) Let \( (A, \preceq) \) be a partially ordered set, and \( C \) be the set of all sequences of elements of \( A \). Define \( \preceq_{1} \) on \( C \) by
\[
\left(a_{n}\right)_{n \in \mathbb{N}} \preceq_{1}\left(b_{n}\right)_{n \in \mathbb{N}} \Leftrightarrow a_{n} \preceq b_{n}: \forall n \in \mathbb{N} .
\]
Show that \( \left(C, \preceq_{1}\right) \) is a partially ordered set.