Hello everyone, my name is Carmen Bruni and today we'll be talking about negations of statements and logic specifically implications. So here we have the following question:
Let $R$, $S$, and $T$ be statements. What is the negation of
$$(\neg R \land S)\Rightarrow \neg T$$In some sense, there's a one line solution, right? We want the negation of this so we can just write
$$\neg((\neg R \land S)\Rightarrow \neg T)$$Then in some sense we'd be done. Sometimes this is all we're asking for, the negation of this thing. However written like this is pretty confusing, it's tough to understand so what I'm going to do is I'm going to try to unpackage this statement. And the way I'm going to do this, I'm going to first start off by getting rid of this implication symbol.
How we do that? we get rid of the implication by saying, say that $(\neg R\land S)$ is statement $A$, and we'll say that $\neg T$ is statement $B$.
The implication, $A\Rightarrow B$, is the same as $(\neg A \lor B)$, logically. So based on that, we will write
$$\neg((\neg R \land S)\Rightarrow \neg T) \equiv \neg(\neg(\neg R\land S)\lor \neg T)$$Now we're gonna bring the negation in
$$\begin{align*} \neg((\neg R \land S)\Rightarrow \neg T) &\equiv \neg(\neg(\neg R\land S)\lor \neg T)\\ &\equiv (\neg R \land S) \land T \end{align*}$$That is all we have to do for this. So what is the negation of $(\neg R \land S)\Rightarrow \neg T$? Well it's $A \land \neg B$, where $A=(\neg R \land S)$ and $B=\neg T$, and that's the negation of any implication, in general. So I wanted to give an example of where we use these logical equivalences, and I wanted to give an example of how something like this might work if you don't want to use, let's say a truth table, or anything like that. You can do this using logical equivalences and things we proved in class.
Alright so maybe I'll just write down the key relationship. The key relationship here was
$$A\Rightarrow B \equiv \neg A \lor B$$Maybe that's just the takeaway. Okay thank you very much for listening.