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**Also sprach Zarathustra** Answers:

1. The first element in $\displaystyle B\cap D$ is 0.

2. $\displaystyle D$ in $\displaystyle <\mathbb{N},\prec_0>$ is the set: $\displaystyle \{0,2,4,6,8,1,3,5,7,9\}$. And 0 is first.

3. I'm trying to prove that the ordered set $\displaystyle <\mathbb{N},\prec_0>$ is well ordered.

So what I have done so far:

Let $\displaystyle A$ be a set of all even numbers. $\displaystyle A\subset \mathbb{N}$, hence $\displaystyle <A,<>$ well ordered.

Let $\displaystyle B$ be a set of all odd numbers. $\displaystyle B\subset \mathbb{N}$, hence $\displaystyle <B,<>$ well ordered.

Let $\displaystyle D$ be non-empty subset of $\displaystyle \mathbb{N}$.

Let we look on $\displaystyle A\cap D$. Suppose $\displaystyle A\cap D \neq \emptyset$. Let $\displaystyle a$ be first element in $\displaystyle A\cap D$, in well ordered set $\displaystyle <A,<>$.

$\displaystyle a$ is first element in $\displaystyle D$ in $\displaystyle <\mathbb{N},\prec_0>$.

Explanation for my last sentence:

For all $\displaystyle b$ element of $\displaystyle D$ which is different from $\displaystyle a$, if $\displaystyle b$ is even, then $\displaystyle b\in A\cap D$, hence $\displaystyle a$ is first.

If $\displaystyle b$ is odd, $\displaystyle a$ is first by definition of order $\displaystyle \prec_0$.

Now, if $\displaystyle A\cap D = \emptyset$ then $\displaystyle B\cap D \neq \emptyset$.

Let $\displaystyle b$ be first element in $\displaystyle B\cap D$ in $\displaystyle <B,<>$.

How can I show that $\displaystyle b$ is first element in $\displaystyle D$ in $\displaystyle <\mathbb{N},\prec_0>$ ?

Thanks again!