Show that the following argument is valid; 1. p $\rightarrow$ q, 2. r $\rightarrow$ s, c. (p $\wedge$ r) $\rightarrow$ (q $\wedge$ s)

Question

So, I've been stuck on this question all day, and the problem I'm having is it just seems intuitive/obvious to me but I'm not sure how to prove it. Anyway, so the full question is;

Show that the following is a valid argument, where the hypothesis appear on lines 1 and 2 and the conclusion appears on line c.

1. p $\rightarrow$ q

2. r $\rightarrow$ s

c. (p $\wedge$ r) $\rightarrow$ (q $\wedge$ s)

All I've done on it so far is simplified different sides, etc and done a couple of truth tables, but not sure what that actually tells me, anyway I got no where, so any help would be GREATLY appreciated.

Thanks :)

Answer 1

What is your inference rule for proving a conditional?

Typically it is some version of: if the temporary assumption $A$ leads to the conclusion $C$, you can infer $A \to C$.

If that's available to you, then you want to make the temporary assumption $(p \land q)$ and aim to prove $(r \land s)$.

How do you do that? As Mauro says in his comments, you extract the conjuncts $p$ and $q$ from the temporary assumption. Now use your two premisses to deduce respectively $r$ and $s$. Conjoin the results and you are home free!

That's the obvious strategy: the exact layout for your proof, what you call the rules of inference in your commentary on the proof, etc., will depend on the fine details of the system of rules of inference you are using.