Tautological Implication of Conditional Statements

Question

Does $P \Rightarrow Q $ and $Q \Rightarrow R$ tautologically imply $(P \land Q)\Rightarrow(Q\land R)$ and $(P\lor Q)\Rightarrow(Q\lor R) $?

\begin{array}{llr} 1. & P \Rightarrow Q & \\ 2. & Q \Rightarrow R & \\ 3. & (P \land Q)\Rightarrow(Q\land R)&TI,2,3 \\ 4. & (P\lor Q)\Rightarrow(Q\lor R) &TI,2,3 \end{array}

Thank you very much for your help in advance!

Answer 1

Yes. Just informally:

For the first one: if $P \land Q$, then $Q$, and thus (given $Q \rightarrow R$) you get $R$. So you have $Q$ and $R$, and so $Q \land R$

For the second: if $P \lor Q$, then either $P$ or $Q$ (or both). If $Q$, then certainly $Q \lor R$, and if $P$, then (given $P \rightarrow Q$) we get $Q$, so again we get $Q \lor R$