Inverse of the negation of the inverse

Question

I'm reviewing some old discrete quizzes in preperation for an upcoming final and would like some insight for the following question:

Given A ⟶ ~B, find the inverse of the negation of the inverse.

Okay. So I can do this some extent, but at the last step I'm not sure how best to proceed.

Firstly, the inverse of A ⟶ ~B would be ~A ⟶ ~~B, or ~A ⟶ ~B, right?

From there, I can take its negation, which should come out to ~A ∧ ~B.

What I have trouble doing is taking a statement like ~A ∧ ~B and turning it into an implication (as the negation rules I have deal with turning implications into statements). Is there a procedure for doing this?

Answer 1

If we are given the implication $p \rightarrow q$

• Its inverse is $\lnot p \rightarrow \lnot q$
• Its converse is $q \rightarrow p$
• The contrapositive is: $(\lnot q \rightarrow \lnot p) \equiv (p\rightarrow q)$

First find the inverse of $A\rightarrow \lnot B\tag {1}$

$$\lnot A \rightarrow \lnot (\lnot B)\equiv \lnot A \rightarrow B\equiv A\lor B\tag{ inverse of 1}$$

Next, Negate the inverse of $(1)$: $$\begin{align} \lnot (A \lor B)\equiv (\lnot A \land \lnot B)\tag{negation of the inverse of (1)}\end{align}$$