This one took me 7 steps (including the premise).
Keep in mind that in order to apply a rule, you need an exact
instance of the rule. That is, you must already have sentences which exactly fit
the pattern of what the rule says you need as justification, and the new
sentence you write down must be an exact instance of what the rule allows you to
derive. (Occasionally Fitch will let you be a little sloppy, but
not often; when in doubt, make sure the pattern in your proof is exactly the same as the
pattern in the rule.)
In particular, when you use vElim, each subproof must show exactly the sentence you want to conclude. For instance, if you need to prove C ⋁ B, it isn't close enough to prove B ⋁ C. The two sentences are tautologically equivalent, true, but they're syntactically distinct, so they don't exactly fit the pattern you need.
6.5 took me 10 steps (9 non-premise steps). 6.6 took me 12 steps (11
The homework problems for this time would be easier if the Distribution equivalences were rules in the formal system! In fact 6.5 and 6.6 amount to proving both directions of one of those equivalences, so if you could use them as rules the proofs would only take one line!
However, the formal system does not have the distribution laws as rules. Instead it has (so far) only the more basic rules we have discussed: the introduction and elimination rules for =, v, and &. So the proofs need to be constructed using only those rules.
Note that ⋁Elim requires a subproof for each disjunct. The other rules don't require subproofs, so you'll want to construct subproofs only if you're using ⋁Elim. (At this point -- later we'll add other rules that use subproofs.) Also it may help to keep in mind that the premises of the argument and anything derived from them in the main proof is available for use in the subproofs.
Took me 6 steps (4 non-premise steps).
If you approach the problem with the right strategic mindset I don't think it
should take that long. Here's how to apply our general strategy:
First step: look at the premises and see whether any of the elimination rules apply. (Answer: no. One premise is an atomic sentence, and the other is a negation.)
Next step: look at the conclusion and think about the introduction rules. The conclusion, ~Cube(c), is a negation, so the rule to think about is negation introduction. Look up the rule if necessary to see how it works.
When you look up negation intro, you see that you assume the opposite of what you want to prove in a subproof, derive a contradiction, and can then get the negation of your starting assumption back out in the main proof.
That seems promising, so give it a try. Now you're in a subproof in which you've assumed Cube(c). You want to derive a contradiction. To do that you will need to use the _|_Intro rule, and in order to use that rule, you need some sentence or other on one line, and the negation of that very same sentence on another line. At this point in the proof you have three sentences you can work with: Cube(c), Cube(b), and ~(Cube(c) & Cube(b)). Can you see a way to use two of these sentences to derive the opposite of the third? If so, then there's your contradiction!
Similar to You Try It #3.
Took me 10 steps (9 non-premise steps). It's easiest to prove ~A and ~B separately, then use &Intro to get the final conclusion.
Took me 10 steps (9 non-premise steps). I needed to use subproofs inside a subproof.