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I am unable to understand the difference between the hypotheses of the following two mathematical statements.

Example 1: The sum of the first $n$ positive integers is $n \dfrac{ (n + 1) }{ 2 } $

Hypothesis: $n$ is a positive integer. (Note that this is implied for the statement to make sense.)

Conclusion: The sum of the first $n$ positive integers is $n \dfrac{ (n + 1) }{ 2 } $ .

$\\$

Example 2: There is a real number $x$ such that $x = 2^{−x}$.

Hypothesis: None, other than your previous knowledge of mathematics.

Conclusion: There is a real number $x$ such that $x = 2^{−x}$.

We know that the hypothesis is defined to be everything that is assumed to be true. Given example $1$, it seemed obvious that the hypothesis for example $2$ would be "there is a real number $x$". After all, example $2$ is assuming that $x$ is a real number.

However, my textbook says that there is no hypothesis for example $2$. Why is this so? The statements of example $1$ and example $2$ seem identical in structure.

Thank you.

EDIT

Now I am convinced that the author made an error.

The problem set contains the following problem.

When $x$ is a real number, the minimum value of $x(x − 1) \ge −1/4$.

Solution

Hypothesis: $x$ is a real number.

Conclusion: The minimum value of $x(x − 1) \ge −1/4$.

EDIT 2

I think I understand the difference now. Example $2$ is saying that there is a specific real number, $x$, which satisfies the following property. In other words, it says definitively that $x$ exists, it is a real number, and it satisfies the following property - there is nothing left to variation. However, example $1$ makes a general claim for any selected number of positive integers. Similarly, example $3$ makes a claim only when a real number is selected. In other words, the other two leave possibility for variation - example $2$ does not.

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The Pointer
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  • What is the point of these examples? I don't understand. They are very weird (and seemingly pointless) –  Nov 23 '16 at 12:41
  • @OpenBall The point of the examples is to show how a proof is structured. – The Pointer Nov 23 '16 at 12:43
  • Are you self-studying? –  Nov 23 '16 at 12:44
  • @OpenBall Indeed. – The Pointer Nov 23 '16 at 12:44
  • Then I advise you to pick up another book. I don't have anything particular in my mind at the moment, but you can certainly learn how to structure a proof by better means / using better resources. –  Nov 23 '16 at 12:46
  • @OpenBall Why? What's wrong with the examples? – The Pointer Nov 23 '16 at 12:47
  • The difference in structure between the two example statements is that one of them says "Whenever you have a number $n$, a certain thing is true no matter what that $n$ is", while the other is "The properties of real numbers and exponentials works in such a way that there is some number $x$ for which a certain thing is true." In other words, one is a "for all" statement, while the other is a "there exists" statement. – Arthur Nov 23 '16 at 13:27
  • @Arthur But why does one being a "for all" statement and the other being a "there exists" statement affect the hypothesis? I would greatly appreciate it if you could please elaborate on this. – The Pointer Nov 24 '16 at 00:51
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    First of all, I think that these formulations represent nit-picking on the highest level. That being said, the first one is easily written as an "if-then" statement: "If $n$ is a natural number, then..." while the second one is not so simple to write like that. The closest I can think of is "If real numbers and exponentiation as we define them exist, then..." And that's how I interpret the difference. – Arthur Nov 24 '16 at 00:59
  • @Arthur Thanks for the assistance - I appreciate it. I will do my best to think of it in terms of "if-then" statements. Hopefully, someone can come along and recommend a more precise way of differentiating such statements (if such a thing even exists). – The Pointer Nov 24 '16 at 01:08
  • @Arthur I think the author may have made an error. See my edit. – The Pointer Nov 24 '16 at 06:00

1 Answers1

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If you want to rewrite Example 1 as an "if___, then___" statement, it must be :

if $n$ is a positive integer, then the sum of the positive itegers up to $n$ is: $\dfrac {n(n+1)} {2}$.

Of course, also in this case we need a "background knowledge" : the theory of natural numbers. Example : the definition of sum.

Regarding Example 2, the "background knowledge" is the theory of real numbers. Of course, this theory assumes that there are real numbers.

Thus, the mathematical statement :

there is a real number $x$ such that $x=2^{−x}$

has not the form "if___, then___".

Mauro ALLEGRANZA
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  • But did you see my edit? There seems to be an inconsistency by the author. The statement assumes that $x$ is a real number, as opposed to an imaginary number. Therefore, the hypothesis seems like it should be as the author wrote in the problem set (the edit). There is an assumption that we are dealing with the real numbers, rather than the imaginary numbers. – The Pointer Nov 24 '16 at 08:26
  • @ThePointer - NO; the second prove that the "equation" $x=2^{-x}$ has a solution in the domain of real numbers. To prove it, it assumes the theory of reals, and thus - obviously - the existence of reals But what he want to prove is the existence of the real (if any) satisfying the said "equation". – Mauro ALLEGRANZA Nov 24 '16 at 08:33
  • I still do not see how there is any difference here. Example $2$ is clearly making the assumption that $x$ is a real number. If you compare Example $2$ with the other examples, there is no clear difference between them. – The Pointer Nov 24 '16 at 12:42