Inferring by resolution refutation

Question

I was going through the MIT course slides, where on page 37 it gives following premises:

$P\rightarrow Q$

$\lnot P \rightarrow R$

And asks to prove

$\lnot Q\rightarrow \lnot R$

by resolution refutation.

I tried first, but then failed. So I created the truth table on this site:

As can be seem in above picture $\lnot Q\rightarrow \lnot R$ is not TRUE for all values of $P,Q$ and $R$, it means it is not valid. So does the problem itself is wrong? Or I am interpreting it wrong.

Answer 1

As you have verified, you cannot prove it.

With the truth assignment $v$ such that :

$v(P)=v(Q)=$f and $v(R)=$t

both premises are evaluated to true while the conclusion is false.

Thus, the conclusion is not consequence of the premises.

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The issue is not that the conclusion "is not TRUE for all values of $P,Q,R$", but that the conclusion is not TRUE in all the rows where both premises are (see 2nd row).