Question

Prove that ((𝑝 → 𝑞) ⋀ ¬𝑞 ) → ¬𝑝 ) it’s tautology without using truth table?

Answer 1

$\left( \left( p\rightarrow q \right) \land \lnot q \right) \rightarrow \lnot p=\left( \left( \lnot p\lor q \right) \land \lnot q \right) \rightarrow \lnot p=\\=\left( \left( \lnot p\land \lnot q \right) \lor \left( q\land \lnot q \right) \right) \rightarrow \lnot p=\\=\left( \lnot p\land \lnot q \right) \rightarrow \lnot p=\lnot \left( \lnot p\land \lnot q \right) \lor \lnot p=p\lor q\lor \lnot p=T$