Well-ordered sets two examples

Question

Say I have the following two sets:

• $\mathbb{R} \cup \{ 0\}$ under < (less than operator)
• $\{(a,b) | a,b \in \mathbb{Z^+} \}$ under $(a,b) \prec (c,d) \iff (a<c) \land (b <d)$.

For $\mathbb{R} \cup \{ 0\}$ I said yes because every subsets will have a least element. For example $\{0\} \subset \mathbb{R} \cup \{ 0\} \text{ and } \{0,1,2\} \subset\mathbb{R} \cup \{ 0\}$, $0$ is the least element in the last subset.

For $\{(a,b) | a,b \in \mathbb{Z^+} \}$ I am saying no because there is not greatest element in each of these subsets.

Looking for some verification to see if my reasoning is correct with these problems.

Answer 1

1. Note for the first example that $\mathbb{R} \cup\{0\}=\mathbb{R}$, which is known to be non well ordered with the usual order given by <. (The subset $(- \infty,0)$ has no least element).
2. For $\{(a,b) | a,b \in \mathbb{Z^+} \}$, note that this is not a total ordered set. For instance, there is no order relation between $(1,2)$ and $(2,1)$, so it cannot be well ordered.