How do I use rules of inference and implication to solve these two disjunctions?

Question

problem http://puu.sh/7RZmi.png

How do I get from 4 to ~J v ~R?

Answer 1

Let's go back to the start:

Try assuming $R \land J$

$(3) R \land J.\;$ ASSUMPTION

Then by simplification, we have

$(4) R$
$(5) J$

From $(1)$, $R\rightarrow V$, so with $(4)$, using modus ponens, we have

$(6) V.$

From $(2)$, $J\rightarrow \lnot V,$ so with $(5)$ we have

$(7) \lnot V$.

You then derive a contradiction by "anding" (6) and (7):

$(8) V \land \lnot V\;$ ($\land$-Intro)

Because the assumption (3) leads to a contradiction (8), you can negate your assumption:

$(9) \lnot (R \land J)$

$(10)\lnot R \lor \lnot J$, by DeMorgan's on $(9)$.

Answer 2

I assume that you have derived step 3 form step 1 using the equivalence between :

$R \supset V$ and $\lnot R \lor V$.

If we are allowed to do this, I suggest you this derivation :

a) contrapose 1), i.e. from $R \supset V$, derive $\lnot V \supset \lnot R$

then use propositional syllogism :

b) from 2) : $J \supset \lnot V$ and a) : $\lnot V \supset \lnot R$, derive : $J \supset \lnot R$.

Finally, apply the equivalence above to get :

c) $\lnot J \lor \lnot R$.