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I am familiar with the tombstone symbol, "$\blacksquare$", that is used to signify the end of a proof. However, it is my understanding that an example isn't technically a proof. For instance, one can't just find an example of a proposition being true and then claim the proposition to be true.

So, I was wondering if there is a symbol that should be used to signify the end of a counterexample?

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    $\begingroup$ Currently your question is only tangentially related to math education, if it is at all. Can you edit it to explain how it is a question about education rather than a mathematical or historical one? Or maybe the question would be better at math.SE instead. $\endgroup$
    – Chris Cunningham
    Commented Apr 10, 2018 at 20:25
  • $\begingroup$ @ChrisCunninham What is the difference between math.SE and math education? $\endgroup$
    – Jordan
    Commented Apr 10, 2018 at 21:18
  • $\begingroup$ @Jordan Math SE is about math, and Math Educators is about teaching math. If you're currently writing a journal article, your question is off-topic in Math Educators; if you're currently writing a textbook or lecture notes, it may be on-topic. $\endgroup$
    – Uwe
    Commented Apr 10, 2018 at 23:09
  • $\begingroup$ @Uwe Ah, I see, I am very off topic then. I will see about removing this question. :) $\endgroup$
    – Jordan
    Commented Apr 10, 2018 at 23:12

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Two answers:

First: These symbols are not standardized. You will find all kinds of "end of ..." symbols, say, black squares, white squares, black triangles, or any symbol that's contained in a dingbat font. Some authors distinguish between the symbols for "end of proof", "end of example", and "end of definition", some authors use the same symbol everywhere, and some authors mark only the end of proofs, but not the end of examples or definitions. In other words: Do what you like, or whatever the publisher's style guide requires.

Second: Technically, a counterexample is a proof of the negation of the assumption. A counterexample for "for every x and y there is a z such that xz = y" is a proof for "not for every x and y there is a z such that xz = y", or, if you push the negation inside, "there exist x and y such that for every z we have xz ≠ y".

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    $\begingroup$ (+1) I would also like to add that I, personally, like to use $\square$ for the end of a proof, example, or counterexample. When I read a paper, I can put a little check in that box when I grok it. :) $\endgroup$
    – Xander Henderson ♦
    Commented Apr 10, 2018 at 16:15
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    $\begingroup$ @Uwe So technically a counterexample is a proof for disproving something? $\endgroup$
    – Jordan
    Commented Apr 10, 2018 at 17:49
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    $\begingroup$ @Jordan Yes. It's a (usually constructive) proof that some property does not hold. $\endgroup$
    – Uwe
    Commented Apr 10, 2018 at 21:14

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