Are positive rational numbers well-ordered?

Positive integers are well ordered but positive rationals are not because for well ordered, every non empty subset must have least element( least element must belong to subset and there is difference between least element and greatest lower bound).

Is the set of rational numbers complete ordered fields?

Then it’s easy to show that sup(an)=inf(bn)=x but yet x is not a rational number therefore the set of rational numbers is not an order complete field.

What is the set of positive rational numbers?

Positive rational numbers refer to rational numbers when their numerators and denominators are both positive or both negative. Examples of positive rational numbers are 3/8, 9/10, -34/-40, etc.

Why is Z not a well-ordered set?

Suppose x∈Z is a smallest element. But x−1Proof by Contradiction, Z is not well-ordered by ≤.

Are non negative integers well-ordered?

The well ordering principle is that every nonempty set of nonnegative integers has a least element.

Is every countable set well-ordered?

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.) A No. We shall show that any well-ordered set S of real numbers is countable by writing S as a countable union of subsets S1,S2,S3,…, each of which is countable.

Why is complex field not ordered?

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Squares are necessarily non-negative in an ordered field. This implies that the complex numbers cannot be ordered since the square of the imaginary unit i is −1.

Which is greater a negative or a positive rational number?

Comparison of Positive and Negative Rational Number is quite obvious i.e. Positive Rational Number is greater than a negative rational number. Every Rational Number represented on a number line is greater than every other rational number represented present to its left.

What is positive rational number and negative rational number?

A number is rational if we can write it as a fraction, where both denominator and numerator are integers and the denominator is a non-zero number….Positive and Negative Rational Numbers.

Positive Rational Numbers	Negative Rational Numbers
All are greater than 0	All are less than 0

Why integers are 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 and y are both negative, and |x| ≤ |y|