# Question about negating implied propositions

I'm negating this proposition: "If you study you will not fail."

I'm using proposition P: "You study" and proposition Q: "You will fail."

The original statement can be written as "$P → ¬Q.$"

My instructor has the negation of this statement like this:

$¬(P → ¬Q) = ¬(¬P \lor ¬Q) = P\land Q$

Why does $¬(P → ¬Q) = ¬(¬P \lor ¬Q)$ ?

• Re-wording: You will study and you will fail... Not too nice to say that to someone ^^ Nov 12, 2014 at 21:56
• @AlexR No, that's not said. It is denied that "you will study and your will fail" is true. Nov 12, 2014 at 22:03
• @DougSpoonwood The negation is $P\wedge Q$ wich is spoken "$P$ and $Q$"... I don't quite understand your objection? Nov 12, 2014 at 22:05

Because $$A\rightarrow B \equiv \lnot A \lor B\tag{1}$$

Think of this as stating: An implication $$A\rightarrow B$$ is true whenever

• $$A$$ is false: $$\;(\lnot A)$$

OR: $$\;\lor$$

• $$B$$ is true: $$\;(B)$$

Hence we have $$\quad \lnot A \lor B$$.

In your case, we have $$\;A = P\;$$ and $$\;B = \lnot Q$$,

So using $$(1)$$ on your proposition: $$\lnot(P \rightarrow \lnot Q) \equiv \lnot (\lnot P \lor \lnot Q)$$ By DeMorgan's, we get $$\lnot \lnot P \land \lnot \lnot Q \equiv (P \land Q)$$

To verify both statements are equivalent, notice that the statements can only be false when both $P,Q$ are false.

I find it more convenient to use: $$[A\implies B]\equiv \neg[A \land \neg B]$$

Then your statement could be restated as: It cannot be that you both study and fail. $$[P\implies \neg Q]\equiv \neg[P\land Q]$$

The negation is that you both study and fail.
$$P\land Q$$

Or equivalently, by De Morgan's Law:

$$\neg[\neg P \lor \neg Q]$$