Supremum Principle: Nonempty Set A & Upper Bound

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Discussion Overview

The discussion revolves around the supremum principle and its application to a specific nonempty set of rational numbers. Participants explore the conditions under which a set is considered upper bounded and whether it possesses a least upper bound, particularly in the context of rational numbers versus real numbers.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant asserts that the set A={x|x∈Q, x²<2} should have a least upper bound according to the supremum principle, yet claims it does not.
  • Another participant challenges the definition of the set, suggesting it should be phrased as x²<2 and argues that there is no theorem guaranteeing a least upper bound for subsets of the rationals Q.
  • A third participant clarifies that the set A, when correctly defined as A={x|x∈Q, x²<2}, does have a least upper bound in the reals, specifically √2, but not in the rationals.
  • One participant emphasizes that the supremum and infimum property does not hold for the set of rational numbers, prompting questions about the definition of rational numbers.
  • Several participants discuss the nature of rational numbers, clarifying that sets like {1,2,3} are indeed sets of rational numbers, but not the complete set of rational numbers.

Areas of Agreement / Disagreement

Participants express disagreement regarding the application of the supremum principle to the set of rational numbers, with some asserting it does not have a least upper bound while others argue it does in the context of real numbers. The discussion remains unresolved regarding the implications of the supremum principle in different number sets.

Contextual Notes

There are limitations in the definitions and assumptions made about the sets discussed, particularly regarding the distinction between rational and real numbers. The discussion also reflects uncertainty about the conditions under which the supremum principle is valid.

andilus
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According to supremum and infimum principle,
nonempty set A={x|x\inQ,x2<2} is upper bounded, so it should have a least upper bound. In fact, it dose not have least upper bound. Why?
When the principle is valid?
 
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Assuming you are talking about the rationals Q, your set isn't even defined in terms of elements of Q. You should phrase it as x2 < 2. Why should it have a least upper bound? There is no theorem stating that a subset of the rationals Q which is bounded above has a least upper bound in Q. In fact, one way to develop the real numbers is to extend them by Dedekind cuts which, effectively, adds all such upper bounds and gives the reals R. Such subsets viewed as subsets of R do have least upper bounds in R.
 
andilus said:
According to supremum and infimum principle,
nonempty set A={x|x\inQ,x2<\sqrt{2}} is upper bounded, so it should have a least upper bound. In fact, it dose not have least upper bound. Why?
When the principle is valid?

First, I suspect you have not written the set correctly. I believe you meant A= \{x | x\in Q, x^2&lt; 2\}. As a set of real numbers, that does have a least upper bound- it is \sqrt{2}. Since that is not rational, if you think of that set as a subset of the rational numbers, in does not have a least upper bound (in the rational numbers). The "supremum and infimum property" does not hold for the set of rational numbers. In fact, it is a "defining property" of the real numbers.
 
HallsofIvy said:
The "supremum and infimum property" does not hold for the set of rational numbers.

what does "the set of rational numbers" mean? Is {1,2,3} a set of rational numbers?
 
andilus said:
what does "the set of rational numbers" mean? Is {1,2,3} a set of rational numbers?

You're taking a course talking about sups and infs and you don't know what the rational numbers are?

{1,2,3} is a set of three positive integers (they are also rational numbers).
{1,2,3,...} would be the set of positive integers
{...,-3,-2,-1,0,1,2...} would be the set of integers
x is a rational number if x can be expressed as the quotient of two integers.
 
andilus said:
what does "the set of rational numbers" mean? Is {1,2,3} a set of rational numbers?
"The set of rational numbers" means the set of all rational numbers. Yes, {1, 2, 3} is a set of rational numbers but not the set of rational numbers.
 

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