You shouldn't be assigning propositional variables to the pieces of the sentences; they should be predicates, so that you can substitute the day of the week.
- "If I take the day off, it either rains or snows."
Let's call this one $$(\forall d)(T (d) \to (R(d) \vee s(d)),$$
where $T(d)$ is "I take take (day) $d$ off", $R(d)$ is "It rains on (day) $d$", and $S(d)$ is "It snows on (day) $d$".
- "I took Tuesday off or I took Thursday off."
This now becomes $T({\rm Tuesday}) \vee T({\rm Thursday})$.
- "It was sunny on Tuesday."
This becomes $S({\rm Tuesday})$.
- "It did not snow on Thursday."
This becomes $\neg s({\rm Thursday})$. Now you can use the rules of inference (including the ones for quantifiers).
(1) $(\forall d)(T (d) \to (R(d) \vee s(d))$ [Assumption]
(2) $T({\rm Tuesday}) \vee T({\rm Thursday})$ [Assumption]
(3) $S({\rm Tuesday})$ [Assumption]
(4) $\neg s({\rm Thursday})$ [Assumption]
We're also missing some key information. It cannot be both rainy and sunny on day $d$, so we have
(5) $(\forall d)(\neg R(d) \vee \neg S(d))$ [Assumption]
Similarly, it cannot be sunny and snow at the same time. Thus
(6) $(\forall d)(\neg s(d) \vee \neg S(d))$ [Assumption]
Now we can use some rules of inference. It was sunny on Tuesday, so we show that it wasn't raining or snowing on Tuesday.
(7) $\neg R({\rm Tuesday}) \vee \neg S({\rm Tuesday})~~~~$ [Universal Instantiation, (5)]
(8) $\neg R({\rm Tuesday})~~~~$ [Disjunctive Syllogism, (7), (3)]
(9) $\neg s({\rm Tuesday}) \vee \neg S({\rm Tuesday})~~~~$ [Universal Instantiation, (6)]
(10) $\neg s({\rm Tuesday})~~~~$ [Disjunctive Syllogism, (9), (3)]
(11) $\neg R({\rm Tuesday}) \wedge \neg s({\rm Tuesday})~~~~$ [Conjunction, (8), (10)]
(12) $\neg (R({\rm Tuesday}) \vee s({\rm Tuesday}))~~~~$ [Logically equivalent to (11)]
Now we show what days you took off.
(13) $T({\rm Tuesday}) \to (R({\rm Tuesday}) \vee s({\rm Tuesday}))~~~~$ [Universal Instantiation, (1)]
(14) $\neg T({\rm Tuesday})~~~~$ [Modus Tollens, (12), (13)]
(15) $T({\rm Thursday})~~~~$ [Disjunctive Syllogism, (14), (2)]
Now you know you took Thursday off. Now to determine the weather on Thursday.
(16) $T({\rm Thursday}) \to (R({\rm Thursday}) \vee s({\rm Thursday}))~~~~$ [Universal Instantiation, (1)]
(17) $R({\rm Thursday}) \vee s({\rm Thursday})~~~~$ [Modus Ponens, (16), (15)]
(18) $R({\rm Thursday})~~~~$ [Disjunctive Inference, (17), (4)]
Thus it rained on Thursday.
Best Answer
Hint: Fill in the blanks.
By (1), we conclude (5) R is __ .
By (5) and (3), we conclude (6) Q is __.
By (6) and (4), we conclude (7) P is __.
By (7) and (2), we conclude the desired result.