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For example, 12|0 = 0

Interesting I can't find this on google anywhere.

Which logical connective is this?

as in this article

http://www.mathblog.dk/strong-induction/

Thomas
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2 Answers2

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In this context, $\mid$ is a relation: $m \mid n$ means that $m$ divides $n$, or equivalently $n$ is a multiple of $m$. This means that there is an integer $\ell$ such that

$$ n = \ell m.$$

If $m$ and $n$ are positive integers with $m \mid n$, then we must have $m \le n$. The symbol "$\mid$" is read divides and is produced by \mid in $\LaTeX$.

Therefore the proposition in the article, "if $n \in \mathbb{N}$, then $12 \mid (n^4 - n^2)$", simply means that $n^4 - n^2$ is divisible by $12$ for each natural number $n$.

However note that the notation in the article is somewhat abusive in that, for instance

$12 \mid (1^4 - 1^2) = 12 \mid (1- 1) = 0 = 0*12$

should instead be something like

$12 \mid (1^4 - 1^2) \iff 12 \mid (1- 1) \iff 12 \mid 0$, which is true since $\overbrace{0}^n = \overbrace{0}^\ell \times \overbrace{12}^m$ (or since $0$ is a multiple of $12$).

Ben
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  • Somewhat abusive? Hm. From the article, it looks like $$12 \mid (1^4 - 1^2) = 12 \mid (1- 1) = 0 = 012$$ is a very confusing shorthand for $$12 \mid (1^4 - 1^2) = 12 \mid (1- 1)$$ $\land$ $$(1- 1) = 0 = 012$$ – MarnixKlooster ReinstateMonica Nov 25 '13 at 05:55
  • @MarnixKlooster, Yes, I am not so sure about this article. @$ $Alfred, I advise you to stick to the source (page 160), from which this example is "shamelessly stolen". By the way the whole book (Richard Hammack's Book of Proof) is available for free online if you're interested. – Ben Nov 25 '13 at 06:13
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In the C programming language, it means the bitwise "or" function. Write your numbers in base 2, and then "or" each corresponding digit.

E.g. 5|3 = 7, 8|3 = 11.

Stephen Montgomery-Smith
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