Answers:

1. The first element in

is 0.

2.

in

is the set:

. And 0 is first.

3. I'm trying to prove that the ordered set

is well ordered.

So what I have done so far:

Let

be a set of all even numbers.

, hence

well ordered.

Let

be a set of all odd numbers.

, hence

well ordered.

Let

be non-empty subset of

.

Let we look on

. Suppose

. Let

be first element in

, in well ordered set

.

is first element in

in

.

Explanation for my last sentence:

For all

element of

which is different from

, if

is even, then

, hence

is first.

If

is odd,

is first by definition of order

.

Now, if

then

.

Let

be first element in

in

.

How can I show that

is first element in

in

?

Thanks again!