Ex: A passenger on an airline qualifies as a frequent flier if the passenger flies more than $25,000$ miles in one year or takes more than $25$ flights during that year.
I started and made up these three propositions knowing $x$ is a passenger:
$P(x)$ = passenger qualifies as a frequent flier
$R(x)$ = passenger flies more than $25,000$ miles in one year
$S(x)$ = passenger takes more than $25$ flights during that year
Not sure how to symbolize $P(x)\; IF\; (R(x)\; OR\; S(x))$
$P(x)$ IF $R(x) \lor $S(x) is equivalent to IF $R(x) \lor S(x)$ THEN $P(x)$, which translates to $R(x) \lor S(x) \to P(x)$ .
Edit: After the edit of the post it seems to now longer be clear how to translate $R(x)$ OR $S(x)$ - this is simply disjunction: $R(x) \lor S(x)$. I see no reason to try and go with a conjuuction or even an exclusive or; to me it seems clear that the or in the sentence equates a simple inclusive logical or, i.e. $\lor$.
Edit 2: After ther was some discussion in the comments about it, you could, as epxlained in my other post, also make the implication a biconditional:
$R(x) \lor S(x) \leftrightarrow P(x)$
This means that a passenger qualifies for a frequent flier if and only if $R(x) \lor S(x)$ holds true, and if this is false, no other condition can make $P(x)$ true. This would make $R(x) \lor S(x)$ a necessary condition. I think, however, that in this not the intended meaning, I could imagine that one could posssibly also qualify for a frequent flier if, for example, one has made a certain amount of flights within the course of a longer time range, without having made it within a year but instead, say, 50.000 miles within the course of two and a half years. In that sense, I rather think that $R(x) \lor S(x)$ is a sufficient condition, but not a necessary one, as proposed b @G. Sassatelli in the comments, and would therefore be translated with a material implication if $\to$ rather than a biconditional iff $\to$.
In either case, if $R(x) \lor S(x)$ is true, $P(x)$ must be true. If that is not the case, the implication simply does not hold: If the antecedent is true but the consequent is false, the implication is false. In my opinion, it is pretty clear from the assignment that if the condition $R(x) \lor S(x)$ is satisfied, $P(x)$ will be true, so I don't understand why @Jared said that "it's possible that the passenger does meet those requirements but is not a frequent flyer", I think this definitely shouldn't happen.
However, the intutitive intepretation of such a sentence would be a more generic one (i.e. a rule-like one), meaning that it is not that well suitable to say $x = passenger$, because then, the formula becomes only true if the current variable assignment happens to assign to x an individual which is a passenger, and can therefore also only be true for one individual.
Instead, the sentence should rather say: For everyone who is a passenger it holds that if he passenger flies more than 25,000 miles in one year or takes more than 25 flights during that year, then he is a frequent flier.
Therefore, I would establish another predicate for passenger, over which you can then quantify.
Since you already chose $P(x)$ for the frequent predicate, let's call $F(x)$ the predicate for passenger ("flier").
The formula would then be:
$\forall x [F(x) \to (R(x) \lor S(x) \to P(x))]$
Meaning: For all x ($\forall x$) which are a passenger ($F(x)$) it holds that ($\to$) if he flies more than 25,000 miles in one year ($R(x)$) or ($\lor$) takes more than 25 flights during that year ($S(x)$), then ($\to$) it is a frequent flier ($P(x)$).
This would reflect the semantics of the sentence better IMO, but you can also stick with the more simple variant I gave above.