Well-ordering principle - Wikipedia
In mathematics, the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element. [1]
3.7: The Well-Ordering Principle - Mathematics LibreTexts
The principle of well-ordering states that \(\overline{S}\) has a smallest element \(z\). Since \(1\in S\), we deduce that \(z\geq2\), which makes \(z-1\geq1\). The minimality of \(z\) implies that \(z-1\notin \overline{S}\).
Every nonempty set of nonnegative integers has a smallest element. This statement is known as The Well Ordering Principle. Do you believe it? Seems sort of obvious, right? But notice how tight it is: it requires a nonempty set—it’s false for the empty set which has no smallest element because it has no elements at all.
The Well-ordering Principle | Brilliant Math & Science Wiki
The well-ordering principle is a property of the positive integers which is equivalent to the statement of the principle of mathematical induction. Every nonempty set S S of non-negative integers contains a least element; there is some integer a a …
1.2: The Well Ordering Principle and Mathematical Induction
We use the well ordering principle to prove the first principle of mathematical induction. Let \(S\) be the set of positive integers containing the integer 1, and the integer \(k+1\) whenever it contains \(k\). Assume also that \(S\) is not the set of all positive integers.
Well-Ordering Principle - ProofWiki
2023年8月9日 · Every non-empty subset of N N has a smallest (or first) element. That is, the relational structure (N, ≤) (N, ≤) on the set of natural numbers N N under the usual ordering ≤ ≤ forms a well-ordered set. This is called the well-ordering principle. The well-ordering principle also holds for N≠0 N ≠ 0:
Well Ordering Principle: Definition, Proof, Application & Examples.
2023年5月4日 · Definition and Explanation of the Well-Ordering Principle. The well-ordering principle states that every non-empty set of positive integers has a least element. This means that given any set of positive integers, there is always a smallest number in that set. To understand this principle better, let's look at an example.
Well-ordering theorem - Wikipedia
In mathematics, the well-ordering theorem, also known as Zermelo's theorem, states that every set can be well-ordered. A set X is well-ordered by a strict total order if every non-empty subset of X has a least element under the ordering.
4.4: The Well-Ordering Principle - Mathematics LibreTexts
2022年4月17日 · To prove the Well-Ordering Principle, consider a proof by contradiction. Suppose \(S\) is a nonempty subset of \(\mathbb{N}\) that does not have a least element. Define the proposition \(P(n):=\) “ \(n\) is not an element of \(S\) " and then use complete induction to prove the result.
To prove that “P (n) is true for all n ∈ N” using the Well Ordering Principle: • Define the set, C, of counterexamples to P being true. Namely, definea C ::= {n ∈ N | P (n) is false} . • Assume for proof by contradiction that C is nonempty. • By the Well Ordering Principle, there will be a smallest element, n, in C.