# What is the meaning of symbol "|"?

I am reading a paper and there is a picture here:

just beneath the "while" word, you would see "|" symbol. What is it? I have searched http://web.uvic.ca/~salam/math_symbols.html and https://www.wikizero.com/en/List_of_logic_symbols

I cannot find it.

• It is 'such that'. Its used in set notation. Commented Apr 1, 2018 at 12:18
• That just means "such that". For instance, $S=\{s\in \mathbb Z \,|\, s\equiv 0 \pmod 3\}$ denotes the set of all integers which are divisible by $3$.
– lulu
Commented Apr 1, 2018 at 12:19
• Thank you for your comments. Commented Apr 1, 2018 at 12:19
• whatever is after that symbol is the property that all elements of the set must follow. There can be multiple properties. Commented Apr 1, 2018 at 12:26
• ... so $C_k$ is constructed by looping over all sets contained in $L_{k-1}$ and adding every possible single object not already contained in that subset to these; but we remove all those sets again that have a $k-1$-element subset not in $L_{k-1}$ Commented Apr 1, 2018 at 12:31

So, if we wanted to talk about the even natural numbers, any of these would work. $$\begin{split} \{ n \in \mathbb{N} \mid n \text{ is even} \}\\ \{ n \in \mathbb{N} : n \text{ is even}\}\\ \{ 2n : n \in \mathbb{N} \}\\ \{ n \in \mathbb{N} \mid (\exists m) [n = 2m]\} \end{split}$$ This is called set builder notation. It is not formally defined, but is a part of informal mathematical language that has developed along with the need to specify sets inside mathematical texts.
The other use of $|$ in logic is to represent the Sheffer stroke, but that is not at all what is suggested in the question above. In programming, $|$ is sometimes used for OR or for bitwise OR. Wikipedia has a longer list of uses in mathematics in their article on "vertical bar".
• Actually, I would use $\{2n\colon n \in\mathbb N\}$ (different spacing). Commented Apr 5, 2018 at 18:48