# Transform several disjunctions to implications

I want to transform a statement which only consists of disjunctions to a statement which only consists of implications.

My statement: j => (a | b | c | d)
Goal: new statement with only implications

My idea:
1. step: j => e | f where e = a | b and f = c | d
2. step: j => (not e => f)
3. step: j => ((not (not a => b)) => (not c => d))

I need for a homework puzzle. But apparently, my solution is wrong.

Do you know what's wrong?

• What is the suggested solution ? I think : $\lnot a \to (\lnot b \to (\lnot c \to d))$ Jan 10, 2019 at 12:13

A different approach to :

$$j \to (a \lor b \lor c \lor d)$$,

applying the equivalence : $$(p \lor q) \equiv (\lnot p \to q)$$ from right to left, leads to :

$$j \to [\lnot a \to ( \lnot b \to (\lnot c \to d))]$$.

Using Exportation we have that : $$\lnot a \to ( \lnot b \to (\lnot c \to d))$$ is equivalent to : $$(\lnot a \land \lnot b) \to (\lnot c \to d)$$.

Now the last step : $$(\lnot p \land \lnot q) \equiv \lnot (\lnot p \to q)$$.

Using it, what we finally get is :

$$j \to (a \lor b \lor c \lor d) \equiv j \to [\lnot a \to ( \lnot b \to (\lnot c \to d))] \equiv j \to [\lnot (\lnot a \to b) \to (\lnot c \to d)]$$.