Maximums and Minimums: Why is the Lower Limit Not 1 in This Set?

  • Thread starter hitmeoff
  • Start date
In summary, the conversation revolves around the concepts of minimum and maximum of a set, as well as the use of set notation. The first example shows a set with a minimum but no maximum because it is explicitly bounded by 0 and Sqrt(2). The second example has no minimum or maximum because the numbers in the set are not restricted to natural numbers, despite the input being restricted to naturals. The third example shows a set with a minimum of 3, as ln 3 is the nearest integer to e^1, but the reasoning for restricting the minimum to integers is still unclear.
  • #1
hitmeoff
261
1
Hello all,

I am currently taking Intro to Real Analysis and we are using Elementary Analysis by Ross. We are on Section 4, dealing with the completeness axiom, but we have only got as far as defining the minimum and maximum of a set (we have not discussed the completeness axiom, archimedean property, upper/lower bounds, supremums/infimums).

We went over Ross' examples in class and I have revied the examples myself and I am confused why the following is so:

1) The set {r in Q: 0 =< r =< Sqrt(2)} has a minimum, namely 0, but no maximum. This is because Sqrt(2) does not belong to the set, but there are rationals in the set arbitrarily close to Sqrt(2).

Ok, I get that. But the next example says:

2) Consider the set {n^(-1^n) : n in N}. This is shorthand for the set {1, 2, 1/3, 4, 1/5, 6, 1/7...} The set has no maximum and no minimum.

Ok, I understand why there is no maximum to the set, there are infinitely many more naturals that this set generates (ie. 8, 10, 12, 14.. -> infinity). What I do not understand is why is the lower limit not 1? 1 is the smallest possible natural number in this set. I know the set continues towards zero getting closer and closer to zero without ever reaching there, but those fractions are not naturals.

If in the first exampl2 Sqrt(2) is not the max because it is not a rational and you can get infinitely many rationals on your way to Sqrt(2) (ie. the max is restricted to rationals) why is the minimum of the second example not 1 (if we are restricting the min to naturals)?

Is it because the set in the first example is explicitly bounded by 1 and Sqrt(2) but the set in the second example is not explicitly limited?
 
Physics news on Phys.org
  • #2
Anyone?
 
  • #3
hitmeoff said:
Hello all,

I am currently taking Intro to Real Analysis and we are using Elementary Analysis by Ross. We are on Section 4, dealing with the completeness axiom, but we have only got as far as defining the minimum and maximum of a set (we have not discussed the completeness axiom, archimedean property, upper/lower bounds, supremums/infimums).

We went over Ross' examples in class and I have revied the examples myself and I am confused why the following is so:

1) The set {r in Q: 0 =< r =< Sqrt(2)} has a minimum, namely 0, but no maximum. This is because Sqrt(2) does not belong to the set, but there are rationals in the set arbitrarily close to Sqrt(2).

Ok, I get that. But the next example says:

2) Consider the set {n^(-1^n) : n in N}. This is shorthand for the set {1, 2, 1/3, 4, 1/5, 6, 1/7...} The set has no maximum and no minimum.

Ok, I understand why there is no maximum to the set, there are infinitely many more naturals that this set generates (ie. 8, 10, 12, 14.. -> infinity). What I do not understand is why is the lower limit not 1? 1 is the smallest possible natural number in this set. I know the set continues towards zero getting closer and closer to zero without ever reaching there, but those fractions are not naturals.
It said "n in N". That does NOT mean that the numbers in the set are in N! The numbers in the set are, as it said, 1, 2, 1/3, etc. It would not say those numbers are in the set if it were intended to restrict the set to the integers.

If in the first exampl2 Sqrt(2) is not the max because it is not a rational and you can get infinitely many rationals on your way to Sqrt(2) (ie. the max is restricted to rationals) why is the minimum of the second example not 1 (if we are restricting the min to naturals)?
Because the first example specifically said that the numbers in the set were natural numbers. The second example said the number, n, used to generate the numbers in the set was restricted to the natural numbers but did NOT say that the numbers in the set had to be natural numbers and the numbers shown as belonging to the set are clearly NOT natural numbers.

Is it because the set in the first example is explicitly bounded by 1 and Sqrt(2) but the set in the second example is not explicitly limited?
No, the first set is clearly has 0 as a lower bound, whether "explicite" or not.
 
  • #4
Ok, I think I get it. Maybe I am getting confused on reading set notation.

So if we say the set S = {r in Q: 0 =< r =< Sqrt(2)} is like saying the set of rationals ranging from [0, Sqrt(2)] In which case 0 is a rational and the smallest value of the set so its the minumum and there is no maximum because there are infinitely many rationals on the way to Sqrt(2).

and

set T = {n^(-1^n) : n in N} is like saying the set of values generated by the function n^(-1^n) with the naturals as input. There is no min or max because there is no minimum or maximum value to the function.

Am I right on this? If I am, then let me point out a third example given by the professor:

{n in Z: 1 =< ln n =< 10} So do I read this "the set of values given by the function ln n, in the range [1, 10], with integers as the input"? So why is the min of this set 3? I know e^1 is 2.7... so the nearest integer is 3, but why are we restricting the min to just integers? obviously n can only be those numbers in the interval [e^1, e^10], I just don't get how 3 is even a member of this set? Is this set not: {ln 3, ln 4, ln 5...ln 22026} (e^10 = 22026.46...}

Or should I read {n in Z: 1 =< ln n =< 10} like "the set of integers n, subject to the function ln n in the interval [1,10]" in which case the integer 3 is the smallest integer that, when subject to ln n, gives a value that is in the interval [1,10]? Would the max of this set then be 22026?
 
Last edited:
  • #5
hitmeoff said:
Or should I read {n in Z: 1 =< ln n =< 10} like "the set of integers n, subject to the function ln n in the interval [1,10]" in which case the integer 3 is the smallest integer that, when subject to ln n, gives a value that is in the interval [1,10]? Would the max of this set then be 22026?

In other words, is this set : {3, 4, 5,...22026}?
 

1. What are maximums and minimums?

Maximums and minimums are the highest and lowest values that a function or set of data can reach within a given range or interval.

2. How are maximums and minimums calculated?

Maximums and minimums can be calculated by finding the critical points of a function and evaluating them to determine which values yield the highest and lowest outputs.

3. What is the significance of maximums and minimums in science?

Maximums and minimums are important in science because they can indicate the most extreme values that a system or process can reach, which can provide valuable insights into its behavior and potential limitations.

4. Can maximums and minimums change over time?

Yes, maximums and minimums can change over time as the underlying function or data set changes. This is particularly relevant in dynamic systems or processes.

5. How can maximums and minimums be used in practical applications?

Maximums and minimums can be used in practical applications to optimize processes and make predictions about future outcomes. For example, in economics, finding the maximum and minimum points of a supply and demand curve can help determine the most efficient pricing strategy.

Similar threads

  • Calculus
Replies
5
Views
1K
Replies
16
Views
2K
  • Classical Physics
3
Replies
85
Views
4K
  • Calculus and Beyond Homework Help
Replies
2
Views
456
Replies
3
Views
1K
Replies
4
Views
187
  • General Math
Replies
5
Views
761
Replies
9
Views
846
Back
Top