Least Upper Bounds: Find, Exist & Belong to Set

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In summary, the homework statement is trying to find a least upper bound and greatest lower bound for a set of equations, but does not mention an "x". The Attempt at a Solution provides a lower bound of 1 for the set, but does not mention an "x". I have decided that 1 is the lower bound for the set.
  • #1
elizaburlap
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Homework Statement



Find the least upper bound and greatest lower bound (if they exist) of the following sets and state whether they belong to the set:

a. {1/n:n[itex]\in[/itex]"Natural Number"}
b. {x[itex]\in[/itex]"Rational Number":0≤x≤√5
c. {x irrational:√2≤x2}
d. {(1/n)+(-1)n:n[itex]\in[/itex]"Natural Number"}

Homework Equations



Not applicable.

The Attempt at a Solution



a. least upper bound does not exist; greatest lower bound is 1 and does belong to the set
b. least upper bound is √5 and does not belong to the set; greatest lower bound is 0 and does belong to the set.
c. least upper bound is 2 and does not belong to the set; greatest lower bound is √2 and does belong to the set.
d. I am not sure about this one, I don't know what the graph would look like.

Am I getting the right idea here? Any ideas for d.?

Thanks!
 
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  • #2
For a. Is 1 <= x, for all x in your set? Revise the definitions of "greatest lower bound", i.e. "infimum".
 
  • #3
Well, I thought it was, because x has to be a natural number.
 
  • #4
elizaburlap said:
Well, I thought it was, because x has to be a natural number.

Unless I'm missing something, n has to be a natural number. Numbers of the form 1/n, where n is natural, are rational numbers.
 
  • #5
elizaburlap said:
Well, I thought it was, because x has to be a natural number.
? There is NO mention of an "x" in problem (a). There is mention of a natural number, n, and the problem talks about the number 1/n for each n:
if n= 1, 1/n= 1
if n= 2, 1/n= 1/2
if n= 3, 1/n= 1/3
if n= 4, 1/n= 1/4
...

I strongly recommend that you write out at least a few of the numbers in each problem.
 
  • #6
This is why I thought that the greatest lower bound was one.

Because n=1, 1/n=1.
as far as I know n cannot be smaller than one as a natural number. Doesn't this make 1 the greatest lower bound??
 
  • #7
elizaburlap said:
This is why I thought that the greatest lower bound was one.

Because n=1, 1/n=1.
as far as I know n cannot be smaller than one as a natural number. Doesn't this make 1 the greatest lower bound??

As Halls mentioned, ##S = \{1,1/2,1/3,\, ...\}##. Plot a few of those points on the ##x## axis. Then remember that a greatest lower bound of a set is at least a lower bound. Then ask yourself if 1 is a lower bound for ##S##.
 
  • #8
Okay! Thanks!
 
  • #9
LCKurtz said:
As Halls mentioned, ##S = \{1,1/2,1/3,\, ...\}##. Plot a few of those points on the ##x## axis. Then remember that a greatest lower bound of a set is at least a lower bound. Then ask yourself if 1 is a lower bound for ##S##.

elizaburlap said:
Okay! Thanks!

To whom are you replying? And what have you decided about this problem?
 

FAQ: Least Upper Bounds: Find, Exist & Belong to Set

What is a least upper bound?

A least upper bound, also known as a supremum, is the smallest number that is greater than or equal to all the numbers in a given set.

How do you find the least upper bound of a set?

The least upper bound of a set can be found by arranging the numbers in the set in ascending order and then selecting the last number in the list.

Does every set have a least upper bound?

No, not every set has a least upper bound. A set must have an upper bound for a least upper bound to exist.

What is the difference between a least upper bound and a maximum?

The least upper bound is the smallest number that is greater than or equal to all the numbers in a set, while the maximum is the largest number in a set.

How is the concept of least upper bounds used in mathematics?

Least upper bounds are commonly used in the study of real analysis, where they are used to prove the existence of limits and to define the completeness of a set of numbers.

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