I agree with the above comment; we have to say that the question must be rephrased as :
show that : "if $p∨(q∨r)$ and $¬q$, therefore $(p∨r)$" is a valid argument.
But as you say in 1) :
$\varphi_1, \varphi_2,\ldots,\varphi_n \vDash \psi$ iff $\vDash \varphi_1 \land \varphi_2 \land \ldots \land \varphi_n \to \psi$.
Thus, the "procedure" of truth-table verification used to establish the validity [i.e. "tautologuesness"] of the conditional it is enough to show the validity of the corersponding argument.
For 2), the definition of valid argument is "formalized" with the relation of logical consequence that, for propositional logic is :
$\Sigma$ tautologically implies $\tau$ (written : $\Sigma \vDash \tau$) iff every
truth assignment for the sentence symbols in $\Sigma$ and $τ$ that satisfies every member of $\Sigma$ also satisfies $τ$.
Here the set of premises $\Sigma$ is : $\{ p∨(q∨r), ¬q \}$, while the conclusion $\tau$ is $(p∨r)$ and the truth table show that in all rows where both premises are true, also the conclusion is.
Thus, the conclusion is tautologically implied by the premises, i.e. the argument is valid.
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
You are confusing "Swimming only if Sunny" with "Swimming if Sunny".
The former is meant by the problem poser to mean "If it is Sunny, we might or might not go Swimming, by we won't go Swimming if it is not Sunny".
The latter, which is how you interpreted it, means "If it is Sunny, we certainly go Swimming; I am saying nothing about what happens if it is not Sunny."
You therefore had the implicatoin in the opposite direction of what was intended.
This matter is further complicated by the fact that $A \mbox{ only if } B$ is often used in math texts to mean $A$ if and only if $B$, as in
"An even number is perfect only if it is of the form $2^{n-1}(2^n-1)$ with $(2^n-1)$ prime."
If you take "only if" to have that meaning, then your confusion is at least half justified.