hey this is PR toer mathematical
induction is a powerful proof technique
involving Two Steps step one prove a
statement is true for the base case
usually Nal 0 or Nal 1 step two prove
that if the statement holds for some
natural number n then the statement also
holds for n +
one this creates a logical Domino which
then proves that the statement holds for
all natural
numbers Let's do an example to prove
that the sum of the first n odd numbers
is equal to
n^2 we start with the base case of the
first odd number it is true that 1 is
equal to 1
2 we now assume the formula is true for
n and then we want to show that it is
also true for n + 1 the next odd number
which will be two larger which will then
be 2 n +
1 how can we prove this formula we'll
also have the same
form we start out with the sum of the
first n odd numbers and we know that is
equal to N squared by the inductive
hypothesis we have the sum of the first
n + one odd numbers is therefore equal
to n^2 + 2 n + 1 we can now Factor n^2 +
2 n + 1 to get n + 1 the quantity
squared and this shows the formula also
holds for the next odd number therefore
we have proven the statement by
induction we'll now get to the thing I
want to show you in this video which is
we will prove that in every group size n
everyone has the same age this will be a
false proof but we're going to proceed
by induction and the challenge for you
is to figure out where the mistake in
this proof
is step one in a group of size n equals
1 there is only one person everyone in
this group trivially has the same age
therefore we have checked the base case
of Nal
1 now we will show whenever the
statement is true for a group of size n
it will also be true for a group of size
n+1 consider a group with n+1 people
that we will number from 1 to n +
1 if we remove person n+1 from the group
then the remaining set of people from 1
to n is a group of n people by the
induction hypothesis everyone in this
group has the same
age similarly if we remove person one
then the remaining set of people from 2
to n + 1 is also a group of n people
once again by the induction hypothesis
everyone in this group has the same
age by steps four and five persons n and
n+1 have the same age as everyone else
in the group numbered from 1 to n minus1
one and therefore they have the same age
as each
other this means everyone in the group
has the same
age by induction we have proved everyone
has the same age for groups of any size
therefore everyone in the world has the
same
age naturally this is an absurd
conclusion so the question is which Step
has the mistake
can you figure it
out give it a moment of thought before
watching the rest of the
video so each step in this proof is
valid except for step
six it is not true that persons n and
n+1 will have the same age even though
they could have the same age as everyone
else in the
group why is that the statement is not
always true with the group of just two
people consider a group with one and
two if we remove person one from the
group then everyone remaining which is
just person two that person does have
the same age and if you remove person
two then everyone remaining which is
just one also has the same
age however
just because everyone else remaining has
the same age does not mean that one and
two will have the same age as each other
person one may have the age of one and
person two may have the age of two so
this is the flaw in this proof that it
will not work when n equals 2 and
therefore the induction domino effect
doesn't happen we can't say it's
therefore then true for three and four
and so on so clearly not everyone in the
world has the same
age induction is a powerful proof
technique but each step has to be valid
or you may end up with an absurd
conclusion did you figure it
out thanks for watching this video
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video please check out my books there
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