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Welcome everybody to the third week of discrete
mathematics.
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In this week, we will be continuing our study
of proof techniques and in particular we will
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look at the proof technique of contradiction
and contrapositiveness.
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So to recall, we were dealing with the techniques
of proving a statement like A implies B.
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There are many proof techniques that can be
applied for proving A implies B, that is constructive
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proof, proof by contradiction, contrapositive,
induction, counter example, existential, etc.
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This is something I have told again and again
in all the video lecture regarding proof techniques,
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that which proof to apply depends on the problem.
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In this course, we will be giving you all
the different proof techniques by list, most
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of them.
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But, whether to, which proof technique to
apply for which problem completely depend
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on you.
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Sometimes the problem can be split into smaller
problems and that can make it easier.
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Sometimes viewing the problem in a different
way can make it easier.
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But, whether to split a problem or how to
split a problem or how to look at in a different
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way is an art that has to be developed by
you.
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In other words, there are some thumb rules
that we will give, for example, if the problem
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is of this kind, then this kind of techniques
can be helpful.
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But at the end of the day, which proof technique
to apply depends fully on your skill that
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you have to develop by doing a lot of practice.
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But till now, we are seeing some of the tricks.
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The first thing that we saw was, how to split
a problem into smaller problems, when the
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B, that is the deduction is of the form of
C AND D. So, in other word B is of the form
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C AND D, then A implies B is same as proving
A implies C AND A implies D.
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We saw an example, how to use this particular
way of splitting the problem can help you
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to split a problem into two parts, we saw
one such example.
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The next technique that you learn was that
sometimes reducing assumption can be helpful.
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In other words, there can be assumptions that
are not necessary, and we can throw them away.
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For example, if I have been asked to prove
A AND C implies B and all that I need is A
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and I can prove A implies B, then that means
that this C is a redundant assumption and
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we can safely throw it away.
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But given the set of assumptions, one has
to find out what are the actual subsidiary
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assumption that might be useful, the rest
of them can be thrown away.
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So in other words, A implies B is good enough,
I mean A implies B implies A AND C implies
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B. Again which assumptions are needed and
which assumptions can be thrown away, depend
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on the problem and you have to identify them
using your intelligence.
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The third technique or third thing that we
learnt was that, sometimes proving something
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harder can actually be easier.
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So in other word, if we have to prove A implies
B and if we can prove that C implies B, then
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A implies B is follows.
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So in other words, while it might be harder
to prove A implies C, but if we can prove
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A implies C, then we get A implies B. Sometimes,
this harder problem technique is easier to
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prove.
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For example, here note that A implies C is
strictly harder than A implies B, but proving
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the harder statement can actually more helpful,
more easier than solving the easiest statement
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A implies B. Now moving on, after the set
of useful tricks we looked at the problem,
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and the first major approach of solving a
problem, namely constructive proof.
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The idea of constructive proof is that if
we have to solve A implies B we work with
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A and start working at it and end up to the
B. Now split up the constructive proof into
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two parts, first part was the direct proof,
meaning directly prove A, meaning directly
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prove B from A. The second one was the case
study, where you split the problems into smaller
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problems depending on A.
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To recall, for direct proof, either you can
start from A and make a step by step deduction
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and end up proving B, or sometimes this direct
proof can be quite magical.
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So we might want to come up with a different
technique, the technique that we suggest it
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was that is going backwards.
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Namely, start with B, we will start with the
thing that you have to prove, work with it
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and slowly simplify it, to get a simpler statement
which might be easier to proof from A.
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So in other words, if you can prove that,
if you can simplify B to C, that means C and
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B are basically equivalence statement, just
that C has been more simplified in that case
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proving A implies B is same as proving A implies
C. So working with B would help you to, working
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with B and simplifying A to C will help you
to understand how to finally prove A implies
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C. Since C is the simplified form, so proving
A implies C will be easier.
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Now, in the context of case study, the idea
was that sometimes we can split up the problem
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into cases depending on A. So in other words,
if the statement is of the following that
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A equals to C OR D, that means C OR D implies
B, then A implies B can be split up as C implies
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B AND D implies B. So sometimes, that we can
split up into cases that A is either C OR
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D and in that case C implies B AND D implies
B.
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We saw a couple of examples, where we use
this case study to solve the problems.
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So this was what we did till now, namely we
looked at some of the techniques of splitting
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the problem into smaller parts and how to
go about attacking that.
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On this video lecture, we will be looking
at a completely new technique which we call
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proof by contradiction where the idea is to
view the problem in a different way.
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So, here is
the idea, the idea is that, note that proving
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A implies B, or this technique A implies B
is equivalent to the statement not B AND A
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is false.
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So in other words, to prove A implies B, one
can prove NOT B AND A is false.
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This is what we call the proof by contradiction.
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So if you have asked that, okay, assume A
and then prove B, then what you were asked
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to prove or what you can prove is this statement
that namely NOT B AND A is false.
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From this particular expression that we have,
a similar statement can be drawn which is
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also of this form, that A implies B is equivalent
to NOT B implies NOT A. This is called the
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proof by contrapositive, and we will be using
this technique after a couple of video.
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So in this video, we will be looking at the
proof by contradiction namely this part, where
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we will be using NOT B AND A is false to prove
some statements.
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It is a very powerful proof technique and
we apply it quite a lot in our various proofs,
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mathematical proofs.
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So what is that we do in the case of proof
by contradiction.
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A good example is to consider this following
example, it is not a mathematical example
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as such but something that is useful.
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Say we want to prove that the earth is not
flat.
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So, how do you prove that earth is not flat?
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One way of doing about it is that, okay, let
us try to prove it directly, then we say,
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okay, when we see a ship coming from the horizon
we see first the top of the ship and slowly
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the complete ship arrives, right.
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And this falls on the usual proof that we
have seen that if I am standing here and there
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is the ship here.
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First what we see is the top of the ship,
and then much later on we see the bottom of
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this ship, the bottom of the ship appears
slowly.
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And the only way that can happen is that,
if it is not flat.
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This is the first technique that one can apply.
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This is what we call as the direct proof technique,
which is the direct proof or the constructive
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proof.
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The second way of saying is that, okay, let
us assume that the earth is flat.
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In that case, there is something weird happens;
we will get a false statement.
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So, if the earth is flat, the ship is coming
from the horizon the whole ship will appear
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at the same time.
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But that does not happen, we see the mast
first, and then the whole ship, and hence
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we get a contradiction.
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Although, these two statements looks so much
similar, but the way in which these were presented
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are very different.
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It is also tells us something important, namely
these proof techniques are not necessarily
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written on stone.
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A single problem can have multiple different
proof techniques.
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Now, all we are saying is that here there
are different kind of proof techniques, and
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you can choose any one of them.
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And for particular problems obtaining a proof
using one proof technique might just be easier
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than obtaining a proof using the other proof
technique.
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And that is all that we are saying in this
different types of proof techniques, right.
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So in other words, proving that the earth
is not flat can be done either using the constructive
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proof or using a proof by contradiction.
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So let us see an example.
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To see the example, you have to consider primes,
and you have to prove that, the primes are
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infinite, in other word there are infinitely
many primes.
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So in other words what you have to prove,
you have to prove is that for all n greater
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than or equal to okay, so let us see an example,
so consider this problem of primes.
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Now what are primes?
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We know primes are numbers that cannot be
divided by any other integer less than itself.
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So what are the prime that we know off?
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The prime that we know of are 2.
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3.
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5.
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7.
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11, 13, 17, 19, 23, and so on and so forth.
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Now is it that there is only a finite the
prime, for example there are only 1000 primes
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or 10,000 primes or something like that or
is it that there is always some prime.
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In other words, we want to prove that there
are infinite number of primes.
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Or in other words, we want to say that the
primes are not bounded by any large number.
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So for all n, integer n, positive integer
n, there always exist a number, which is bigger
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than n and which is prime, so there is a prime
that is always bigger than some number, so
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you can pick your favourite n, so you tell
me, okay.
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Is there aprime bigger than 10 lakhs and I
should be able to be produce you one.
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So this proves that the number of prime are
not bounded by a large enough integer, they
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are infinite.
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Then how you prove this statement, how do
you prove that the number of primes is indeed
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infinite.
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So to prove that we will prove it using contradiction.
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So first of all, let us go back one step what
is the, if you want to formulate this statement,
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that prove primes are infinite, if I am formulating
in terms of A implies B and what is B and
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what is A. So the B is here.
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The B is
primes are infinite and what is A and as such
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no A here.
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So many times, you will get such kind of questions.
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A in this case is actually everything that
we know to be correct everything we know to
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be correct.
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So in fact it basically says that with all
the knowledge that you have can you prove
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primes are infinite.
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So if I have to prove this statement, using
the contradiction how will we get about it.
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So one way of going about it is, ofcourse,
remember, recall we have to prove that this
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whole thing is convert into not B and A. This
is false, right.
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Or In other words, this statement says that
B is not if number of primes is not infinite
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or in other words if the number of primes
are finite, then something that we know must
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be contradicted.
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So There is something so A hereis something
that we know must have a contradiction, right
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and this is how we will go about proving our
statement.
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So let us continue with the proof.
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So let us assume that there are finitely many
primes.
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So that means, there is a largest prime, let
us call that one pt and in that case, we say
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that the set of total primes is p1 to pt,
p1, p2 till pt.
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Now consider this number the product of all
these primes plus 1.
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Now, what is this number?
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First of all, note that this number is strictly
bigger than pt, why?
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If you remember that we know that p1 is 2,
we know p2 is 3 and so on.
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So this number p1 times p2 times dot, dot,
dot, till pt is bigger than twice pt or 6pt
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and so on, right, so that number, this number
product of all the pt plus 1.
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If this 1 turns out to be a prime, then what
happens?
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Then it contradicts the fact first of all
that pt is the largest prime.
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We have assumed pt is the largest prime, so
it contradicts the statement pt is largest
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prime, because this one is clearly greater
than pt.
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Now can or is the product of p1 to pt plus
1 a prime?
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Now if the product plus 1 is not a prime,
then by the definition of a prime, something
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must be dividing it and when something divides
it, we have looked at this proof earlier,
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that is some number integer divides another
integer, then there must be a prime that divides
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the other integer.
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So in other words from one of p1 to pt must
divide this sum because p1 to pt are the only
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set of primes that we have.
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But all the primes p1 to pt divides, of course
the product of them, so when we divide the
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product plus 1, this number, by any number,
say, does p1 divides this number, when you
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divide this number by p1, what is the remainder?
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The remainder is 1, similarly when I divide
by p2, the remainder is 1, similarly when
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I divide by the prime pt, the remainder is
1 or in other words, none of this primes p1
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to pt divides this new number.
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So this number is not divisible by any of
the old primes p1 to pt.
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What does it mean?
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It means that, there is only one that this
number has to be a prime.
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There is no number, no prime can divide this
number hence this one is a prime and this
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is the contradiction as we have discussed
earlier, because now we have a number that
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is a prime that is bigger than the largest
prime, which cannot be.
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So this proves that our initial assumption
of pt being the largest prime, is false.
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Or In other words, it cannot be a largest
prime or in other words, pt cannot be, or
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number of primes cannot be finite.
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So basically, we proved that if p1 to pt are
the set of primes, finite set of primes, then
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I created a new number which was larger than
largest prime and we prove that it has to
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be prime, which is a contradiction.
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So this is a typical proof by contradiction
where we start from assuming that the statement
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that we have to prove is not true and we worked
our way through and at the end proved that
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it contradicts something, either it contradicts
what you are assuming or it contradicts something
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that we know and so on and so forth.
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There are many related problems to this particular
problem and I leave it to you as an exercise
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for you want to prove that there are infinitely
many primes of the form 1(mod 4).
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So namely what are the primes 1(mod 4), say,
5, 13, 17, 29 and so on.
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Prove that there are infinitely many primes
of the form 1(mod 4).
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00:26:35,899 --> 00:26:45,399
Similarly prove that there are infinitely
many primes of the form 3(mod 4).
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00:26:45,399 --> 00:26:52,049
Again, similarly prove that there are infinitely
many primes of the form 1(mod 6) and there
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are infinitely many primes of the form 5(mod
6).
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00:26:58,399 --> 00:27:11,132
So we will be doing some more problems using
contradiction in the next video.
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So in the next video, we will be proving some
statements like that square root 2 is not
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00:27:24,320 --> 00:27:25,320
a rational number.
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00:27:25,320 --> 00:27:26,999
What is a rational number?
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00:27:26,999 --> 00:27:34,330
A real number is rational if it can be written
as ratio of two integers.
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00:27:34,330 --> 00:27:39,220
So if a number can be written as p by q where
p and q are integers, then we say it is is
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a rational number and 1, 2, 3 they all can
be written as 1/1, 2/1, 3/1, 2/3, 49/99 and
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so on.
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00:27:52,450 --> 00:27:53,510
We claim that square root 2 is not rational
number.
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In the next video, we will be proving this
particular problem using the same contradiction
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technique.
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I encourage you guys to go and try to solve
this problem by yourself before you see the
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00:28:11,029 --> 00:28:12,029
next video.
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Thank you.