1
00:00:00,000 --> 00:00:04,000
Hello everyone, my name is Carmen Bruni
and today we'll be talking about

2
00:00:04,000 --> 00:00:08,000
negations of statements and logic

3
00:00:08,000 --> 00:00:10,000
specifically implications.

4
00:00:10,000 --> 00:00:14,600
So here we have the following question: 
"let R, S, and T be statements,

5
00:00:14,600 --> 00:00:17,566
What is the negation of the 
following compound statement:

6
00:00:17,566 --> 00:00:23,400
'not' R 'and' S 
implies 'not' T.

7
00:00:23,400 --> 00:00:26,433
In order to solve this…
well I mean, in some sense 

8
00:00:26,433 --> 00:00:28,433
there's a one-line 
solution, right?

9
00:00:28,433 --> 00:00:32,000
We want the negation of this
so we can just write 'not',

10
00:00:32,000 --> 00:00:34,533
and then copy our
statement problem.

11
00:00:34,533 --> 00:00:34,566


12
00:00:34,566 --> 00:00:36,866
And then in some 
sense we'd be done.

13
00:00:36,866 --> 00:00:40,866
Sometimes this is all we're asking 
for, the negation of this thing.

14
00:00:40,866 --> 00:00:44,900
However, written like this it's pretty 
confusing, and tough to understand,

15
00:00:44,900 --> 00:00:47,466
so what I'm going to 
do is I'm going to try to 

16
00:00:47,466 --> 00:00:49,733
  'unpackage'
this statement.

17
00:00:49,733 --> 00:00:53,266
And the way I'm gonna do this, I'm
going to first start off by getting rid

18
00:00:53,266 --> 00:00:56,200
of this implication symbol.

19
00:00:56,200 --> 00:01:01,166
And how we do that, we get rid 
of the implication by saying…

20
00:01:01,166 --> 00:01:04,600
call this A: 
" 'not' R 'and' S"

21
00:01:04,600 --> 00:01:06,933
we'll call that statement A and 
we'll call this statement B

22
00:01:06,933 --> 00:01:10,233
The implication, A implies B, 
         is the same as

23
00:01:10,233 --> 00:01:12,466
'not' A 'or' B,
logically.

24
00:01:12,466 --> 00:01:16,266
So based on that, 
we will write,

25
00:01:16,266 --> 00:01:24,266
'not' A which is 'not' 
bracket 'not' R 'and' S

26
00:01:24,266 --> 00:01:25,733


27
00:01:25,733 --> 00:01:27,000
'or'

28
00:01:27,000 --> 00:01:28,300


29
00:01:28,300 --> 00:01:32,600
'not' T, so the B part.

30
00:01:32,600 --> 00:01:33,966
There it is. Okay.

31
00:01:33,966 --> 00:01:38,166
So then there's 'not' A
 'or' B, which is 'not' T.

32
00:01:38,166 --> 00:01:41,766
Now we're gonna 
bring the negation in.

33
00:01:41,766 --> 00:01:45,766
And doing that so the
'not not' will cancel.

34
00:01:45,766 --> 00:01:49,000
Then we'll be left with...

35
00:01:49,000 --> 00:01:53,433
'not' R 'and' S.

36
00:01:53,433 --> 00:01:53,566


37
00:01:53,566 --> 00:01:58,366
And then we have - the 'not' will pass to
the 'or' which will give us an 'and'

38
00:01:58,366 --> 00:02:01,300
and then the 'not' 
will pass to the 'not' T.

39
00:02:01,300 --> 00:02:02,633
and that will 
give us a T.

40
00:02:02,633 --> 00:02:06,866
And...that is all
we have to do for this.

41
00:02:06,866 --> 00:02:09,800
So what is the negation
 of this statement?

42
00:02:09,800 --> 00:02:13,933
Well it's "A
and not B", 

43
00:02:13,933 --> 00:02:18,866
and that's the negation of any
implication, in general.

44
00:02:18,866 --> 00:02:22,800
So I wanted to give an example of 
where we use these logical equivalences,

45
00:02:22,800 --> 00:02:25,900
and I wanted to give an example of 
how something like this might work

46
00:02:25,900 --> 00:02:28,766
if you don't want to use, let's say a 
truth table, or anything like that.

47
00:02:28,766 --> 00:02:32,033
You can do this using logical
equivalences and things we proved in
class.

48
00:02:32,033 --> 00:02:36,600
Alright so maybe I'll just write 
down the key relationship.

49
00:02:36,600 --> 00:02:36,966


50
00:02:36,966 --> 00:02:40,033


51
00:02:40,033 --> 00:02:42,733
"The key relationship was

52
00:02:42,733 --> 00:02:51,000
A implies B is equivalent 
to 'not' A 'or' B"

53
00:02:51,000 --> 00:02:51,033


54
00:02:51,033 --> 00:02:53,600
Maybe that's just 
the takeaway.

55
00:02:53,600 --> 00:02:57,933
Okay thank you
very much for listening.