How to prove an inference is valid given two premises?

Question

I am new to discrete math, and I am stuck in a proof that asks me to infer $q \land r$ from the premises $p \land q$ and $p \to r$.

My attempt is first to change $p \to r$ into $\neg p \lor r$ and do the same for inferring $p \land q$ and $p \to r$ to $q \land r$ . But then my proof is stuck. Can anyone give me some advice on how to proceed from here? Or should I not have started this way?

Thank you!

Answer 1

$p$ is true (from $p\land q$), and $p \implies r$, so $r$ is true (modus ponens). Because $q$ is also true (from $p \land q$), then $r$ and $q$ are true, in other words, $q \land r$.

Answer 2

Using the equivalent form of implication, then the distributive property: $$\require{cancel}(p \land q) \land (p \to r) \;=\; (p \land q) \land (\lnot p \lor r) \;=\; \cancel{(p \land q \land \lnot p)} \lor (p \land q \land r) \;\;\implies\;\; q \land r$$