Inductive Reasoning (Part - 3) - Mathematics, Engineering Video Lecture | Logical Reasoning for UGC NET

Luis looked at the expression x
minus y times x squared plus
xy plus y squared, and
wrote the following.
He said, using the distributive
property-- so he
took x minus y and he
distributed this expression
onto the x and this expression
onto the negative y.
So that's why we have an
x times this entire
expression over here.
x times the entire expression
minus y times this entire
expression.
Then he says, using the
distributive property again,
this is equal to-- so then he
distributed this x into that
and got all of the first
three terms here.
He got these first three
terms. And then he
distributed the y.
He distributed the y over here
and he got these three terms.
And then he saw, looks like
that this term x squared y
cancels out with a minus x
squared y, and the xy squared
cancels out with the negative
xy squared.
Then he saw that it equals
x to the third
minus y to the third.
And he wrote down, I conjecture
that x minus y
times x squared plus xy plus y
squared is equal to x to the
third minus y to the third
for all x and y.
Did Carlos use inductive
reasoning?
Explain.
Well, inductive reasoning is
looking at a sample of
things-- looking at some set
of data, if you will-- and
then making a generalization
based on that.
You're not 100% sure, but based
on what you've seen so
far, you think that the pattern
would continue.
Or you think it might be true
for all things that have that
type of property or whatever.
Now in this situation,
he didn't look at
some type of a sample.
He actually just did a proof.
He multiplied this out
algebraically.
In fact, it's incorrect for him
to say I conjecture here.
A conjecture is a statement or
proposition that is unproven,
but it's probably going
to be true.
It's unproven but it seems
reasonable, or it seems likely
that it's true.
This isn't a conjecture.
This is proven.
He proved that x minus y times
x squared plus xy plus y
squared is equal to x to the
third minus y third.
He should have written-- and
this is a much stronger thing
to say-- he should have said, I
proved that this is true for
all x and y.
So to answer the question, did
he use inductive reasoning?
No.
I would say that he made
an outright proof.
No, he made a proof.
Inductive reasoning would have
been, if he would have saw, if
you would have given him 5 minus
2 is equal to-- or 5
minus 2 times 5 squared plus 5
times 2 plus 2 squared, and
you saw that that was equal to
the same thing as 5 to the
third minus 2 to the third.
And then let's say he did it
for, I don't know, one in
seven in a couple of examples.
And it kept holding for all the
examples that it was the
first number cubed minus the
second number cubed.
Then it would have been
inductive reasoning to say
that that is true for
all numbers x and y.
But here it's not
an induction.
He didn't use induction.
Or I shouldn't say induction.
He didn't use inductive
reasoning.
He outright proved that
this statement is true
for all x and y.