Can a colon be used in predicate logic?

Question

Is a colon or letter $e$ used to describe if something (lets say $x$) is an element or part of something? What's the correct way to write it out?? My lecturer showed us to write it using a colon but another video on YouTube showed me to use the letter $e$.

For example, is it:

$\exists x:S, \neg ( (D(x) \wedge c(x) )$

or:

$\exists xeS, \neg ( (D(x) \wedge c(x) )$

Thanks!

Answer 1

The little 'e' is non-standard. It's sometimes used as a poor substitute for the "element of" relation $\in$, when typing mathematics without Latex (or extended characters). I cannot overstate how shoddy this is considered to be; never, under any other circumstances, use it in lieu of $\in$ (and even in those circumstances, I would do everything in my power, including typing the words "is an element of", to avoid using 'e' to replace $\in$). Certainly, when you're writing by hand, or on this site, you should use the $\in$ character. It can be accessed through Latex/MathJax by typing \in.

The colon is something different. It usually stands for "such that". You could write, $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : x + y = 0,$$ which reads,

For all $x$ in the set $\mathbb{R}$, there exists a $y$ in the set $\mathbb{R}$ such that $x + y = 0$.

It's not about something being an element of something else, it's a symbol short for the phrase "such that". Formally, it's not really a necessary phrase; it arguably makes the logical symbols a bit more readable. You may, if you wish, omit the comma and colon, and write $$(\forall x \in \mathbb{R})(\exists y \in \mathbb{R})(x + y = 0).$$ It means the same thing.