Math Problem Statement
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
Solution
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:
- What is the difference between a well-ordered set and a totally ordered set?
- How does the well-ordering principle relate to mathematical induction?
- Can you explain how the set of integers is ordered compared to natural numbers?
- What are some examples of sets that are not well-ordered?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Set Theory
Well-Ordering Principle
Natural Numbers
Real Numbers
Formulas
-
Theorems
Well-Ordering Principle
Suitable Grade Level
Grades 9-12
Related Recommendation
Proving the Well-Ordering Principle Using the Principle of Mathematical Induction
Well-Ordering of Finite Sets: Proving Minimal Elements in Ordered Sets
Understanding the Truth of the Statement: Vx ∈ N, x > 0
Natural Number System: Explained (1, 2, 3, ...)
Natural Numbers as a Subset of Integers: True or False