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 the following compound statement:
'not' R 'and' S implies 'not' T.
In order to solve this… well I mean, in some sense
there's a one-line solution, right?
We want the negation of this so we can just write 'not',
and then copy our statement problem.
And 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 it's pretty confusing, and 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 gonna do this, I'm going to first start off by getting rid
of this implication symbol.
And how we do that, we get rid of the implication by saying…
call this A: " 'not' R 'and' S"
we'll call that statement A and we'll call this statement B
The implication, A implies B, is the same as
'not' A 'or' B, logically.
So based on that, we will write,
'not' A which is 'not' bracket 'not' R 'and' S
'or'
'not' T, so the B part.
There it is. Okay.
So then there's 'not' A 'or' B, which is 'not' T.
Now we're gonna bring the negation in.
And doing that so the 'not not' will cancel.
Then we'll be left with...
'not' R 'and' S.
And then we have - the 'not' will pass to the 'or' which will give us an 'and'
and then the 'not' will pass to the 'not' T.
and that will give us a T.
And...that is all we have to do for this.
So what is the negation of this statement?
Well it's "A and not B",
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 was
A implies B is equivalent to 'not' A 'or' B"
Maybe that's just the takeaway.
Okay thank you very much for listening.