Welcome everybody to the last video lecture
in week three. This week, we have looked at
various proof techniques and we will continue
to look at more proof technique in this particular
video lecture.
So till now we have seen that we have rather
talked about that to prove A implies B, there
can be many proof techniques, namely Constructive
Proof, proof by Contradiction, proof by Contrapositive,
induction, counter example and Existential
Proof. Till now we have seen constructive
proofs, proof by contradiction and proof by
Contrapositive.
Now here I again repeat that is repeated in
every of this video lectures, namely which
proof to apply for which problem is something
that you have to decide for yourself. Some
of the problems can be split into smaller
problems that can be easier to tackle while
some of them can be viewed in a different
way and by viewing so one can make the problem
easier. But which problem to split and how
to split it or how to look at it is an art
in itself that we have to develop.
In this particular set of video lectures,
we will be giving you thumb rules on which
one to use, which proof technique use for
which problem, but at the end of the day you
will have to make a choice for yourself. Many
of the problems can have multiple different
proof techniques that is good enough for it.
We also looked at the some of the simple techniques
of how to split the problems into smaller
cases.
Particularly, we looked at the following two
techniques, first of all if B can be written
as C AND D then proving A implies D is same
as proving A implies C AND A implies D. And
in this case one can split the problem of
proving A implies B into this two parts A
implies C AND A implies D. There is also the
other thing of removing the redundant assumptions,
in other words sometimes some assumptions
are there which are not necessary.
Say in other words says if I have been asked
to prove A AND C implies B, one might throw
away the C and end up proving A implies B,
which is good enough for us. So in other words
what I am saying is that, one should extract
the most relevant set of assumptions that
are necessary to prove B. That would simplify
the proof or rather that will simplify the
problems and help in getting the proof easier.
The third part was that sometimes proving
something stronger is easier. So we might
have something like C implies B and we have
to proof A implies B, it might be easier to
prove A implies C instead, which would be
a harder problem in itself, but might be easier
to prove. So in that case, making the problem
harder can make our life of getting a proof
is easier. Other than these three tricks,
we also solve some proof technique.
In particularly, we looked at constructive
proofs and in that we saw direct proof. In
direct proof the idea is that to prove A implies
B, we start with the assumption A and step
by step go on to prove B. But sometimes by
doing so or doing such a thing the proof can
became very magical and hence one can imagine
that it is harder to get such a proof. Another
technique that can be there is what is known
as a backward proof.
So in this technique, to prove A implies B,
the idea is to simplify B slowly till the
time it is simplified to a different form,
so namely if I simplify B to C then A implies
B would basically mean A implies C. And since
A implies C is a simpler statement, it will
be easier to get that proof. So this was the
Constructive proof and in which case we saw
the direct proof technique, there was the
other proof technique in the Constructive
proof, which is called the case studies.
The idea is that sometimes the assumptions
or the premise can be split into different
cases. And in that case we can split the problem
according to cases namely if A can be written
as C OR D then A implies B is same as proving
C implies B AND D implies B. Here it is important
to note that how to split A into this two
things C OR D, is something that require some
amount of art and understanding of the problem.
One would like to split up into C OR D in
such a way that proving C implies B AND D
implies B becomes easier.
We have of course seen example at all the
various problem case studies that we have
seen or various proof techniques that we have
seen till now. Now the third one that we saw
the proof technique was the proof by contradiction,
the idea was that to prove A implies B one
can also end up proving NOT B AND A is false,
so this is called the proof by contradiction.
So there we assume that B is not true and
you work our ways till we get some real statement
which is always false.
A very similar statement to this same technique
is what is known as the proof by contrapositive,
there the idea is that proving A implies B
is same as proving NOT B implies NOT A and
sometimes proving NOT B implies NOT A can
be easier to prove.
So in particular, if B can be written in the
form of C OR D, in that case A implies B,
which is same as NOT B implies NOT A is same
as NOT C AND NOT D implies NOT A. So this
form of NOT C and NOT D implies NOT A is sometimes
easier to prove. One thing notice that, all
of these things are similar, all the proof
techniques are similar in the sense that they
all are different ways of writing A implies
B.
How all problems can be solved with any of
these proof techniques? But it so happens
that some of the proof technique is easier
to get or work with, for certain problems
and that is what we are trying to tell you.
Now this was what we have done till the last
week.
In this particular video, we will look at
the case when the statement that we have asked
to prove is actually false. For example we
can get some problem like prove or disprove
A implies B. And let us assume that this statement
is actually false, that means A does not implies
B. So if A does not implies B or if A implies
B is not true then what you will do? A thing
to note here is that a statement is not true
is for some setting of variables to true or
false the statement is false.
Here can I somehow put the variables the true
and false and get some false statement. So
a something like the true implies false kind
of statement. So in other words, we have to
prove that a NOT of A implies B is true for
some instance, where there is an instance
for which we can see that A implies B is not
true. So A implies B is not true or otherwise
NOT of A implies B is true.
Now to prove that NOT of A implies B is true
for some instance, how do you go about it.
So usually the problems are of the form FOR
ALL x or their exist x something happens.
So say the problem is of the form FOR ALL
x, prove that A(x) implies B(x). So the negation
of this one as we have seen is that, it is
their exist x and A(x) implies B(x) opposing,
or in other words A(x) does not implies B(x).
Now what we mean by A(x) does not implies
B(x)? Recall that A implies B is same as B
ORNOT A.
So that means the negation of the A(x) implies
B(x), or that means A(x) does not implies
B(x) is same as NOT of B(x) OR NOT of A(x).
So NOT of the B(x) OR NOT of A(x), which is
the like De Morgan’s law there exist x as
NOT of B(x) AND A(x). So in other words we
have to produce an x, such that B(x) does
not hold but A(x) holds, right. So this is
what we have to prove. So to disprove in other
words disprove a statement of the form A implies
B, which is the form there exist A(x) implies
B(x) one has to produce a x such that B(x)
does not hold and A(x) holds.
So to prove the original statement is not
true, we have to find an x such that this
statement is true. And this is what we call
proof by counter example, this is one of the
few cases where an example can give you a
proof. So we will see a couple of examples,
problems in this case.
So look at the first problem, this is a problem
that we started in the first video only. So
this problem says that to prove or disprove
for all positive integer n, n square minus
n plus 41 is prime. Now how to disprove it?
The way to disprove it is to produce an n,
such that n square minus n plus 41 is not
a prime. If this statement is not true and
we have to find an n such that this number
is not a prime.
Now to producing such an n is not necessarily
easier job here. For example, one can try
to prove that or one can check that, if I
put n equals to one or two or three or four
or five or so on, it will always turn out
to be a prime. The lowest n for which this
number is not a prime is unfortunately as
high as 41. So only when we put 41 n equals
to 41, we realize that this number is not
prime.
Thus we disprove the statement by demonstrating
a counter example here which is n equals to
41. Finding this counter example is not at
all easy and sometime depending on the problem
can take a lot of hard work. In fact, there
are problems for which the counter example
has been found after centuries of hard work.
So one such example is, say, prove or disprove
for all positive integers n, 2 power 2 power
n plus 1 is a prime. Now these are what are
known as the Fermat primes, Fermat after the
famous French mathematician Fermat. He was
there over a century ago and he has post this
problem. Unfortunately for n equals to 1 it
is not that hard to calculate as you can see
it comes to 3 which is prime.
For n equals to 1 it is 5 which is a prime,
n equals to 2, this number again the, 2 power
2 power n plus 1 becomes 17, which is also
a prime. n equals to 3 it is again a prime,
n equals to 4 now we already can see the things
are becoming very complicated, here it is
2 power 2 power n in other words 2 power 2
power 4 plus 1, which stands out to be 65537
is a prime. Now, what about n equals to 5?
And what it turns out is that n equals to
5 is horrendously big number 4294967297, which
is actually not a prime, because it is a product
of these numbers. As you can see or imagine
first of all proving that this number is not
a prime, is not at all an easy job because
it’s prime factorization is quite complicated.
So, to prove or disprove this statement whether
2 power 2 power n plus 1 is a prime is not
necessarily easy, but sometimes this is the
only way of proving or disproving some statements.
Thus, what we can say is that, to disprove
a statement one can do so by giving an instance
where the statement fails. We can prove, we
call this thing proof by counter example and
finding a counter example can be very hard
and require both ingenuity and sometimes very
high computational powers. So this demonstrates
one more example of proof technique, namely
counter example, a very useful proof technique
for proving or disproving a statement.
Note that proving a statement does not, is
not necessarily the easiest job and neither
is disproving a statement. In both of the
cases, we need proper proof techniques. Till
now we have looked at, the proof by Construction
or constructive proof and proof by Contradiction,
Contrapositive and proof by Counter example.
In the next week, we will continue our study
of proof techniques by looking at proof by
induction, an extremely powerful proof technique
that we will be looking at. Thank you.
