Then by definition, all subsets of Z has a smallest element. ... But x−1<x, which contradicts the supposition that x∈Z is a smallest element. Hence there can be no such smallest element. So by Proof by Contradiction , Z is not well-ordered by ≤.
Is Z an ordered set?
By definition, any well-ordered set is totally ordered . However, the converse is not true – the set of integers Z, which is totally ordered, is not well-ordered under the standard ordering (since Z itself and some its subsets do not have least elements). Although, any finite totally ordered set is well-ordered.
Why are integers not well-ordered?
Integers. Unlike the standard ordering ≤ of the natural numbers, the standard ordering ≤ of the integers is not a well ordering, since, for example, the set of negative integers does not contain a least element . ... x is positive, and y is negative. x and y are both positive, and x ≤ y.
How do you prove well-ordered?
An ordered set is said to be well-ordered if each and every nonempty subset has a smallest or least element . So the well-ordering principle is the following statement: Every nonempty subset S S S of the positive integers has a least element.
What is well-ordered set example?
An example of a well-ordered set is the naturally ordered set of natural numbers . On the other hand, the interval of real numbers [0,1] with the natural order is not well-ordered. ... A totally ordered set is well-ordered if and only if it contains no subset that is anti-isomorphic to the set of natural numbers.
Is Z+ totally ordered set?
The unique order on the empty set, ∅, is a total order . ... The set of real numbers ordered by the usual “less than or equal to” (≤) or “greater than or equal to” (≥) relations is totally ordered, and hence so are the subsets of natural numbers, integers, and rational numbers.
Is Q well-ordered?
A set T of real numbers is said to be well-ordered if every nonempty subset of T has a smallest element. Therefore, according to the principle of well-ordering, N is well-ordered. Show that Q is not well-ordered .
Are the rationals well-ordered?
The rationals, for example, do not form a well-ordering under the usual less-than relation, but there is a way of putting them into one-to-one correspondence with the natural numbers, so it can be well-ordered by the total order implied by this correspondence. Any countable set can be well-ordered.
Can every set be ordered?
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.
Is an empty set well-ordered?
Note that every well ordered set is totally ordered , and that if X is empty, then the unique (empty) ordering on X is a well ordering.
What does having a well-ordered Day mean?
1 : having an orderly procedure or arrangement a well-ordered household .
How do you prove division algorithms?
1 (Division Algorithm). Let a and b be two integers with b > 0. Then there exist unique integers q, r such that a = qb + r, where 0 ≤ r<b. The integer q is called the quotient and r, the remainder.
Can a well-ordered set be infinite?
Every finite set is well-ordered. The classic example of an infinite well-ordered set is { 1,2,3,...} , which is infinite but of course only countable.)
Is every well-ordered set well founded?
In order theory, a partial order is called well-founded if the corresponding strict order is a well-founded relation. If the order is a total order then it is called a well-order. In set theory, a set x is called a well-founded set if the set membership relation is well-founded on the transitive closure of x.
Is a total order a well Order?
A totally ordered set in which every non-empty subset has a minimum element is called well-ordered. A finite set with a total order is well-ordered. All total orderings of a finite set are, in a sense, the same.
Is the poset totally ordered?
The poset (N,≤) is a totally ordered set . The poset ({1,5,25,125},∣) is also a totally ordered set. Its Hasse diagram is shown below. It is clear that the Hasse diagram of any totally ordered set will look like the one displayed above.
