The Statements are

1) Some Pencils are Pens
2) No Pen is an Eraser
3) All Sharpeners are Erasers

And the Conclusions are

1) No Eraser is a Pencil
2) All Pencils can never be Sharpeners

Which of the conclusions follow according to the statements?


closed as off-topic by 2012rcampion, Deusovi Sep 18 '16 at 14:25

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  • $\begingroup$ Welcome to Puzzling! This seems to be a homework question, which is off-topic here. $\endgroup$ – Deusovi Sep 18 '16 at 14:25
  • $\begingroup$ @2012rcampion it's certainly an interesting question, interesting enough to be worthy of being on PSE, whether or not you think it's some maths problem. $\endgroup$ – Buffer Over Read Sep 18 '16 at 14:32
  • $\begingroup$ @Deusovi No, It's not a home work question, at all. Moreover I'm not a student. I tried to solve a question, which I encountered online(a local forum) but unable to solve it, since I didn't understand the actual meaning of second conclusion. So, I posted it here to know that. I don't think it's a mathematical problem. $\endgroup$ – Omkar Reddy Sep 18 '16 at 14:33
  • $\begingroup$ @Ganesh.R: Many first-year discrete mathematics studentshave homework problems that are exactly like these questions. I myself tutored one a few months ago and there were at least a half dozen of this type. $\endgroup$ – Deusovi Sep 18 '16 at 14:34
  • $\begingroup$ @Deusovi You might be right. But I didn't have this type of questions in my subjects whether it is mathematics or something else. And as I already told you i am not a student. This is not a homework question. $\endgroup$ – Omkar Reddy Sep 18 '16 at 14:44

Rewriting the assumptions in a more formal form:

  1. $\{pencils\}\cap\{pens\}\neq\emptyset$
  2. $(pen)\Rightarrow(not\,\,eraser)$
  3. $(sharpener)\Rightarrow(eraser)$

Now let's look at the two possible conclusions.

  • Conclusion 2 is unclearly phrased. If it means that "the statement that all pencils are sharpeners is untrue", then it does follow from the assumptions, by the following argument.

    Assume all pencils are sharpeners. Then by assumption 3, all pencils are erasers. So by assumption 1, some pens are erasers. This contradicts assumption 2.

    If it means that "no pencils are sharpeners", then it doesn't necessarily follow, because some non-pen pencils could still be sharpeners without contradicting the three given assumptions.

  • As for conclusion 1, it too doesn't necessarily follow from the given assumptions. It would be fine for some eraser to be a pencil, just as long as it's not also a pen.

For an example of a scenario in which conclusion 1 and the second interpretation of conclusion 2 don't follow from the three assumptions, see the following Venn diagram:

Venn diagram

As you can see, in this model some pencils are pens, no pens are erasers, and all sharpeners are erasers, but some pencils are sharpeners and some erasers are pens.


Conclusion 1

does not follow. Perhaps there are exactly two objects: both are Pencils, one is a Pen (but not an Eraser) and the other an Eraser (but not a Pen). It doesn't matter whether they're Sharpeners for this purpose.

Conclusion 2

is ambiguous: does it mean "it can never be the case that all Pencils are Sharpeners", or "for all Pencils p, p can never be a Sharpener"?
The first version of this does follow, for some Pencils are Pens by statement 1, and therefore not erasers by statement 2, and therefore not sharpeners by statement 3.
The second version, though, does not follow; take the same scenario as for conclusion 1, and add that the non-Pen is also a Sharpener and hence an Eraser.

  • $\begingroup$ Nice! Less than 1 minute's difference :-) $\endgroup$ – Rand al'Thor Sep 18 '16 at 13:57
  • $\begingroup$ Yup. But I got there first, so there :-). $\endgroup$ – Gareth McCaughan Sep 18 '16 at 13:57
  • $\begingroup$ (Also, for the conclusion(s) that don't hold I gave actual model(s) of the theory in which the conclusion(s) fail, which I think is the clearest-cut way to prove consistency results.) $\endgroup$ – Gareth McCaughan Sep 18 '16 at 13:59
  • $\begingroup$ I added a Venn diagram - beat that! :-P (And I didn't read your answer first to see what models you'd used.) $\endgroup$ – Rand al'Thor Sep 18 '16 at 14:08
  • $\begingroup$ Now, now children, don't fight - you can both have a cookie! (Or a +1 anyway.) $\endgroup$ – YowE3K Sep 18 '16 at 21:17

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