Supremum Principle: Nonempty Set A & Upper Bound

The set of rational numbers includes all numbers that can be expressed as a ratio of two integers, including fractions, decimals, and whole numbers.
  • #1
andilus
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According to supremum and infimum principle,
nonempty set A={x|x[tex]\in[/tex]Q,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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  • #2
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.
 
  • #3
andilus said:
According to supremum and infimum principle,
nonempty set A={x|x[tex]\in[/tex]Q,x2<[tex]\sqrt{2}[/tex]} 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 [itex]A= \{x | x\in Q, x^2< 2\}[/itex]. As a set of real numbers, that does have a least upper bound- it is [itex]\sqrt{2}[/itex]. 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.
 
  • #4
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?
 
  • #5
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.
 
  • #6
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.
 

FAQ: Supremum Principle: Nonempty Set A & Upper Bound

What is the Supremum Principle?

The Supremum Principle is a fundamental concept in mathematics that states that every nonempty set A of real numbers that is bounded above has a least upper bound, also known as the supremum.

What is a nonempty set?

A nonempty set is a set that contains at least one element. In other words, it is not empty or has a size greater than zero.

What does it mean for a set to be bounded above?

A set is bounded above if there exists a number that is larger than or equal to all the elements in the set. This number is called the upper bound.

What is the least upper bound?

The least upper bound, also known as the supremum, is the smallest number that is larger than or equal to all the elements in a nonempty set A that is bounded above. It is denoted by sup(A).

Why is the Supremum Principle important?

The Supremum Principle is important because it provides a way to define and characterize the largest element in a set. It is used in various areas of mathematics, including analysis and topology, and is a key concept in understanding limits and continuity.

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