Dm Discrete Mathematics Se Cse It The Well Ordering Principle Youtube In mathematics, the well ordering principle states that every non empty subset of nonnegative integers contains a least element. [1] . 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 in s s such that a≤b a ≤ b for all b b ’s belonging.
The Well Ordering Principle The well ordering principle. every nonempty subset of \(\mathbb{n}\) has a smallest element. proof. in fact, we cannot prove the principle of well ordering with just the familiar properties that the natural numbers satisfy under addition and multiplication. hence, we shall regard the principle of well ordering as an axiom. 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. So, the well ordering principle captures something special about the nonnegative integers. while the well ordering principle may seem obvious, it’s hard to see offhand why it is useful. but in fact, it provides one of the most important proof rules in discrete mathematics. 7 sources. theorem. every non empty subset of n has a smallest (or first) element. that is, the relational structure (n, ≤) on the set of natural numbers n under the usual ordering ≤ forms a well ordered set. this is called the well ordering principle. corollary. the well ordering principle also holds for n ≠ 0:.
Comments are closed.