Question

1.    Determine whether this proposition is a tautology.

[(p → q) ∧ (q → r)] → (p→ r) 

Answer 1

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

The formula is a tautology