I'm not a maths natural so I'm not sure if this is a silly question. I was wondering, is it acceptable to structure an expression involving logical quantifiers without using function notation?
For example, can I write
$\exists a \in A f(a) < 3$
or do I need to define $P(a)$ etc. first and write like so:
$\exists a \in A P(a)$ where $P(a) = (f(a) < 3)$ or something weird like that (not quite sure how to write what $P(a)$ is)?
The first option is much shorter but I don't know if it is valid. If it is valid, is it acceptable to show as above or should I additionally put brackets around the predicate part with a space in between?
E.g. $\exists a \in A \text{ } (f(a) < 3)$
Well..., $~\exists x{\in}A~(x=3)~$ is a well-formed-formula that claims "Something in $A$, is equal to three."
This is equivalent to just saying: "Three is in $A$." $$3\in A\quad\equiv\quad\exists x{\in}A~(x=3)$$
You are not required to provide an alias for the predicate, though you may if you wish to do so. Where $P(x)$ is an alias for $(x=3)$ you may substitute. It should be noted in the margin if not made clear nearby in the surrounding text.$$\exists x{\in}A~P(x)\tag{where $P(x):= (x=3)$}$$
PS: Yes: spacing and brackets do help readability.