Math Problem Statement
Use the PMI to prove the well ordering principle. Hint: let S ⊂ N be
nonempty, and define a property P (n) of natural numbers by setting P (n) to be true
exactly when 1, . . . , n̸ ∈ S.
Solution
To prove the Well-Ordering Principle (WOP) using Principle of Mathematical Induction (PMI), we'll follow the hint provided. The Well-Ordering Principle states that every non-empty subset of the natural numbers contains a least element.
Step-by-Step Proof:
1. Setup the Problem:
Let be a non-empty subset of the natural numbers. Our goal is to show that has a least element.
We define a property for a natural number as follows:
- is true if and only if none of the numbers are elements of , i.e., In other words, asserts that the first natural numbers do not belong to the set .
2. Base Case:
We use the contrapositive approach, where we assume is non-empty and attempt to derive a contradiction.
Since is non-empty, we know there must exist some natural number .
Suppose, for contradiction, that has no least element. This would mean that for all , the first natural numbers do not belong to , implying that holds for all .
Start with , which states that the number 1 does not belong to . Since is non-empty, we know must contain some element, but implies .
3. Induction Hypothesis:
Now, assume for some , that is true. This means , i.e., none of the numbers from 1 to belong to .
4. Inductive Step:
We now prove that holds, assuming is true. By , we know that none of the numbers belong to .
We must show that . If , then would be a member of and would be the smallest member of , since by the induction hypothesis, no number less than is in . But this contradicts the assumption that has no least element.
Therefore, , and is true.
5. Conclusion by Induction:
By the principle of mathematical induction, is true for all . This means that none of the natural numbers belong to .
However, this contradicts the fact that is non-empty (since must contain some natural number). Therefore, our original assumption that has no least element must be false.
6. Conclusion:
Hence, must have a least element. This proves the Well-Ordering Principle using the Principle of Mathematical Induction.
Would you like more details or have any questions on specific steps?
Here are 5 related questions:
- How does the contrapositive method support the proof of the well-ordering principle?
- What is the significance of the inductive hypothesis in this proof?
- Can you provide an example of a non-empty subset of natural numbers that has a least element?
- How does this proof compare with direct proofs of the well-ordering principle?
- How can the well-ordering principle be applied in other mathematical contexts, such as divisibility?
Tip: The Well-Ordering Principle and PMI are closely related; many proofs can be structured by converting one into the other. Understanding this connection helps in abstract problem-solving across different areas of mathematics.
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Math Problem Analysis
Mathematical Concepts
Principle of Mathematical Induction (PMI)
Well-Ordering Principle
Set Theory
Formulas
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Theorems
Well-Ordering Principle
Principle of Mathematical Induction
Suitable Grade Level
Undergraduate Mathematics or Advanced High School