Given $(p \land q)$, use the Fitch system to prove $(q \lor r)$

Question

As stated in the title the problem is the following

Given $(p \land q)$, use the Fitch system to prove $(q \lor r)$

Here's my resolution:

    1.  
        p & q
 Premise
    2.  
        p
 And Elimination: 1
    3.  
        q
 And Elimination: 1
    4.  
            r
 Assumption
    5.  
            q | r
 Or Introduction: 3

Is it correct?

Answer 1

You haven't actually asked a question; you're supposed to actually ask a question before you can receive an answer. But I'm going to assume that your question is, "Have I correctly proved what I was asked to prove?"

The answer is, this is not a correct proof of the given statement. You have proved q | r given p & q and r as premises, but you were asked to prove q | r given only p & q as a premise.

In step 4, you assumed r, but this assumption is unnecessary. See if you can prove the statement again without assuming r.

Answer 2

It is the correct idea, but you do not need to raise an assumption to use Or-Introduction.

If you have $\Sigma\vdash q$ then you can infer $\Sigma\vdash q\vee r$ by Or-Introduction.

If you have $\Sigma\vdash p\wedge q$ then you can infer $\Sigma\vdash q$ by And-Elimination.

And since your premise, $\Sigma$, is $p\wedge q$, then you may do just that, in a few lines of a fitch proof.

Try it again.