Proceed methodically: Suppose the premisses are true and conclusion false. So
$1.\quad p \quad \Rightarrow \quad T$
$2.\quad p \lor q \quad \Rightarrow \quad T$
$3.\quad q \to (r \to s) \quad \Rightarrow \quad T$
$4.\quad t \to r \quad \Rightarrow \quad T$
$5.\quad \neg s \to \neg t \quad \Rightarrow \quad F$
From the last, you know
$6.\quad \neg s \quad \Rightarrow \quad T$
$7.\quad \neg t \quad \Rightarrow \quad F$
Whence
$8.\quad s \quad \Rightarrow \quad F$
$9.\quad t \quad \Rightarrow \quad T$
4 and 9 give us
$10.\quad r \quad \Rightarrow \quad T$
So $r \to s$ is false, and hence, from (3)
$11.\quad q \quad \Rightarrow \quad F$.
So we've worked backwards to successfully find a valuation of all the variables (at lines 1, 8--11) which you can check makes all of 1 to 5 true, i.e. makes the premisses true and conclusion false.
Systematizing this "working backwards" method gives us the user-friendly method of "semantic tableaux" or "truth-trees" used in many textbooks (including mine, and Paul Teller's which is freely available online).
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.
Best Answer
The instructions seem clear enough. Construct a table like so and then fill in the missing details.$${\begin{array}{r|l:l}1 & p\to(q\wedge r) & \text{Premise 1 (Hypothesis)}\\ 2 & \neg q & \text{Premise 2 (Hypothesis)}\\\hdashline 3 & \quad p& \text{Assumption (Hypothesis)}\\ \vdots & \quad \vdots& \vdots\\ \underline\quad & \quad \bot& \underline{\quad},\underline{\quad},\textit{rule of inference}\\ \underline\quad & \neg p & 3,\underline\quad,\text{Hypothesis negation}\\ \hline\end{array}\\ \therefore\quad p\to(q\wedge r), \neg q \vdash \neg p}$$