Is the Maximum Always the Same as the Supremum in an Open Interval?

[hammonjj]

Homework Statement

For each subset of ℝ, give its supremum and maximum, if they exist. Otherwise, write none.

Homework Equations

d) (0,4)

The Attempt at a Solution

For part d, if the problem were [0,4], both the supremum and maximum would be 4, since the interval includes the end points, but I'm not sure about when it doesn't. It seems to me that the supremum would still be 4, as that qualifies as a least upper bound because 4 is in the neighborhood of the set, but is the maximum also 4? I don't think it would be since the interval does not include end points, but I'm confused!

Thanks!

 

[LCKurtz]

Homework Statement

For each subset of ℝ, give its supremum and maximum, if they exist. Otherwise, write none.

Homework Equations

d) (0,4)

The Attempt at a Solution

For part d, if the problem were [0,4], both the supremum and maximum would be 4, since the interval includes the end points, but I'm not sure about when it doesn't. It seems to me that the supremum would still be 4, as that qualifies as a least upper bound because 4 is in the neighborhood of the set, but is the maximum also 4? I don't think it would be since the interval does not include end points, but I'm confused!

Thanks!

You aren't too confused. You are correct that the max is not 4. Does the max even exist?

 

[hammonjj]

You aren't too confused. You are correct that the max is not 4. Does the max even exist?

I think it does, but I don't know what to call it. Since my interval is a subset of the real numbers, there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set.

Am I making this too complicated? Does the max have to be some integer?

 

[HallsofIvy]

I think it does, but I don't know what to call it. Since my interval is a subset of the real numbers, there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set.

Am I making this too complicated? Does the max have to be some integer?

No, you are not making it too complicated, nor does the maximum have to be an integer. However, what you are saying, "there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set" is wrong. The maximum of a set is, by definition the largest number in that set. If the set were "[0, 4]" or "(0, 4]" then the maximum would be 4. But (0, 4) is the set of all numbers larger than 0 and less than 4. If "x" is in that set, what could you say about (x+ 4)/2?

 

[hammonjj]

No, you are not making it too complicated, nor does the maximum have to be an integer. However, what you are saying, "there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set" is wrong. The maximum of a set is, by definition the largest number in that set. If the set were "[0, 4]" or "(0, 4]" then the maximum would be 4. But (0, 4) is the set of all numbers larger than 0 and less than 4. If "x" is in that set, what could you say about (x+ 4)/2?

Since x is in the interval (0,4), that is x is less than 4, (x+4)/2 is in that interval.

 

[LCKurtz]

No, you are not making it too complicated, nor does the maximum have to be an integer. However, what you are saying, "there's some x=4-ε, where x is the maximum and ε is some tiny interval that, when subtracted from 4, gives you the maximum of the set" is wrong. The maximum of a set is, by definition the largest number in that set. If the set were "[0, 4]" or "(0, 4]" then the maximum would be 4. But (0, 4) is the set of all numbers larger than 0 and less than 4. If "x" is in that set, what could you say about (x+ 4)/2?

Since x is in the interval (0,4), that is x is less than 4, (x+4)/2 is in that interval.

Even better, it is between x and 4. Hall's point is that any number x you might use for the maximum has to be in the interval so it is less than 4. Yet (x+4)/2 is in the interval and greater than x. So...?