Formally prove the following equations using propositional logic

Question

p ∧ ¬p $\vdash$ q ∧ ¬q

p ∧ r ⇒ q ∧ r , p ∨ r ⇒ q ∨ r $\vdash$ p ⇒ q

I have literally been trying to figure these out all morning and I'm desperately stuck now. We have to prove them using propositional logic via the formal proof/natural deduction system. If anyone could help me out and explain how they reach their conclusion I'd be immeasurably grateful - thanks.

The ruleset used for this include:

and-elimination and-introduction or-elimination or-introduction <=>-introduction <=>-elimination =>-introduction =>-elimination ¬-introduction ¬-elimination

Answer 1

Hint

For the second one, assume $p$ and derive $p \lor r$ by $\lor$-intro.

Then derive $q \lor r$ from the second premise.

Now we have to consider $q \lor r$ for $\lor$-elim.

From the "alternative" $q$, we have immediately $p \to q$.

From the alternative $r$ we get $p \land r$ and thus $q \land r$ from the first premise.

Now apply $\land$-elim to derive $q$ and then $p \to q$ again.

Thus, we may conclude by $\lor$-elim.