What is the meaning of | operator?

Question

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/

Answer 1

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 $12$).

Answer 2

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.