Sorry if this question is too elementary than usual for this site, but I'm trying to analyze the logical form of the following statement:

3 is a common divisor of 6, 9, and 15.

I'm not sure how to go about constructing its logical form. I'm assuming I only have to look at its meaning and leaving the superfluous details out. But when I tried constructing it, it didn't feel or sound right, so I tried looking up a solution.

Let T = 3 is a common divisor, S = 3 is a common divisor of 6, N = 3 is a common divisor of 9, F = 3 is a common divisor of 15

This is what I got: [(T $\wedge$ S) $\wedge$ (T $\wedge$ N) $\wedge$ (T $\wedge$ F)]

I'm assuming one of my mistakes was introducing an auxiliary element (i.e T) that shouldn't have been there. This is an exercise in Velleman's How To Prove it, specifically chapter 1, number 2(c). I tried looking up a solution since there wasn't one in the appendix, and I found this. As you can see, my solution seems to be way off.

QUESTION: Is this correct? And if so, what does the solution mean? (i.e what do the terms represent? what does "6 mod 3 = 0" mean?) I'm still in the beginning of the book so I haven't seen/been introduced to some of these terms, and it came as quite a surprise.

Also, you'll have to forgive my LaTeX noobiness. To be fair, I never even heard of it until I came across this site, so I'm still learning. Feel free to edit my question. Any help would be appreciated (regarding the problem, not LaTeX) thank you! =)

  • $\begingroup$ The format "$a$ is a common divisor of $b$" is not right. By a common divisor is meant a number which divides two or more other numbers, at least as the term "common divisor" is used. $\endgroup$ – coffeemath Mar 8 '13 at 15:56
  • $\begingroup$ And the statement "3 is a common divisor" is just complete nonsense. $\endgroup$ – Chris Eagle Mar 8 '13 at 15:59
  • $\begingroup$ Sorry, but I was never the best math student. I'm just trying to learn. $\endgroup$ – random maths student Mar 8 '13 at 16:00
  • $\begingroup$ Well, at this stage it is rather a linguistic problem. "My grandfather is a common ancestor of my uncle, my cousin and me" makes sense. "My grandfather is a common ancestor of my uncle and also a common ancestor of my cousin and also a common ancestor of me" makes no sense. $\endgroup$ – Hagen von Eitzen Mar 8 '13 at 16:33
  • $\begingroup$ @HagenvonEitzen: Right. $\endgroup$ – random maths student Mar 8 '13 at 16:40

Your main problem, I think, is not understanding exactly what is meant by the statement that $3$ is a common divisor of $6,9$, and $15$. This simply means that $3$ is a divisor of $6$, $3$ is a divisor of $9$, and $3$ is a divisor of $15$. And when you paraphrase it that way, its structure becomes very evident: it’s just $S\land N\land F$, where $S,N$, and $F$ are the statements ‘$3$ is a divisor of $6$’, ‘$3$ is a divisor of $9$’, and ‘$3$ is a divisor of $15$’, respectively.

You could go a little further and notice that these three statements themselves all have the same basic structure. If $T(n)$ is the ‘open’ statement ‘$3$ is a divisor of $n$’, then your original sentence has the logical form $T(6)\land T(9)\land T(15)$.

  • $\begingroup$ This is a very intuitive explanation for me. thanks for clearing up the confusion. And you are probably right.. I think I ended up being concerned about the structure and disregarded the meaning of what I was writing, so I wrote something nonsensical like "T = 3 is a common divisor". $\endgroup$ – random maths student Mar 8 '13 at 16:44
  • $\begingroup$ @randommathsstudent: You’re welcome. $\endgroup$ – Brian M. Scott Mar 8 '13 at 16:52

$a \mod b = c$ means exist $p\in \mathbb{N}$ such that $a = p*b +c$. So $6 \mod 3 =0$ exactly means $3$ is a divisor of $6$.

For your answer, your $T$ as no meaning "3 is a common divisor" of what? But $S,N,F$ or ok. Remark that $S\equiv 6 \mod 3 =0$, $N\equiv 9 \mod 3 =0$ and $F\equiv 15 \mod 3 =0$.

So the answer $S\wedge N \wedge F$ is correct.


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