3

In page 9 of Mathematical Analysis I by Vladimir A. Zorich, I saw the idiom "prove by example":

enter image description here

As you can see, it is completely different from what is described in this post:

Is there such a thing as proof by example (not counter example)

But when I search in Google, did not find any definition for that. Now I want to know is that a standard idiom? If yes, what is the meaning of that?

Community
  • 1
hasanghaforian
  • 465
  • 5
  • 16

3 Answers3

6

I'm not an English native ... but I think that the meaning is simply :

"We will not prove all the stated theorems but, in order to show the method of proof (by way of example), we will prove one of them."

Similar : for example, by way of illustration.

Mauro ALLEGRANZA
  • 94,169
  • That may well be Zorich's intent, but then instead of "prove the first [...] by way of example" it should be "prove these by way of example, showing only the first." – Danny Rorabaugh Oct 08 '16 at 22:14
  • @DannyRorabaugh: your proposed rewording is not correct. "By way of example" is not about a method of proof but about the author's decision to present a proof of only one statement in the theorem. – Rob Arthan Oct 09 '16 at 02:11
  • I agree. My point is that if you attach the prepositional phrase "by way of example" to the verb "prove" when the direct object is "one of these" then it is being treated as a proof technique. By making the direct object "all of these" then "by way of example" makes sense because there is a set from which to take an example. – Danny Rorabaugh Oct 09 '16 at 19:16
4

This is more of a general grammatical point.

Doing something "by way of example" means that you are using it as an example of a typical way of doing something. For example in this case, there are two equalities to be proved, and the two proofs are very similar. Therefore they show you the first proof, and leave the second as an exercise.

preferred_anon
  • 17,121
0

I use "proof by example" to mean that the fact is demonstrated with an example in such a way that makes the method of the general proof clear. For example, if the proof of a number theoretic statement involves a process depending on the prime factorization, walking through the process with, say, $1500 = 2^2\cdot 3 \cdot 5^3$ would allow a clear demonstration of what to do with primes from each class modulo 4 and with various powers.

I would not have used that phrase for this example, where (as Mark S. noted) it was proof by picking an arbitrary element.

Danny Rorabaugh
  • 551
  • 3
  • 15
  • 1
    See the answers by Daniel and Mauro. "By way of example" just means "as an example' (of one the things to be proved). – Rob Arthan Oct 08 '16 at 19:29