The set of all ______ is an example of a set that satisfies the well-ordering principle.

a. positive real numbers c. natural numbers

b. negative integers d. real numbers

The correct answer is:

**c. natural numbers**

The well-ordering principle states that every non-empty subset of the set of natural numbers has a least element. This means the set of natural numbers is well-ordered because every subset of natural numbers will have a smallest number. 

Here’s why the other options do not satisfy the well-ordering principle:

- **a. Positive real numbers:** The set of positive real numbers is not well-ordered because there is no least element (for example, you can always find a smaller number than any given positive real number).
- **b. Negative integers:** While you can have a "least negative integer" in a specific set, the entire set of negative integers doesn't satisfy the principle as there is no smallest negative integer (you can always subtract 1 to get a smaller one).
- **d. Real numbers:** The set of real numbers is not well-ordered, since there is no smallest real number, and subsets of real numbers like the open interval (0, 1) do not have a least element.

Would you like more details on this concept or any specific clarification?

### Relative Questions:
1. What is the difference between a well-ordered set and a totally ordered set?
2. How does the well-ordering principle relate to mathematical induction?
3. Can you explain how the set of integers is ordered compared to natural numbers?
4. What are some examples of sets that are not well-ordered?
5. How does the concept of least upper bounds differ from the well-ordering principle?

### Tip:
The well-ordering principle is equivalent to the principle of mathematical induction and is foundational to many areas of number theory.

The set of all ______ is an example of a set that satisfies the well-ordering principle.
a. positive real numbers
b. negative integers
c. natural numbers
d. real numbers

Discover why the set of natural numbers satisfies the well-ordering principle, a fundamental concept in set theory. This page explains why natural numbers have a least element, unlike sets of real or negative integers. Suitable for Grades 9-12.

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