In this video we will look at a couple of examples to practice thinking about conjectures and counter examples.
Example A says: here's an algebraic equation and a table of values for n with the results for t.
Notice that we started out with this equation:
And what we have here is really just a table,
that's divided in three parts, we have our values for n, all sort of calculations in the middle, and our answer fot t.
And after looking at the table, Pablo makes this conjecture:
the value of (n-1)(n-2)(n-3), in other words, the answer for t is zero for any hole number value of n.
So, he is basically saying, that no matter what number I plug in over on the left from n,
my answer is always going to end up being zero
because that's what happen in the first three times. So maybe it will always happen.
The question that we have is: is this a valid, or true, conjecture?
So, if it's true, it means that it would be true for any number that you plug in for n.
So you can plug in 100, and the answer should still be zero for t.
So let's just test it out. Let's just try 100. Let's say n=100.
We are trying to see if t will actually be equal zero. So let's plug it in.
We would have (100-1)(100-2)(100-3).
100-1 is 99..
Then we've got times 98,
and times 97.
Now I know that the answer to that is not zero,
because to have an answer zero you have to multiply some of the one of the line by zero.
So, this number is not equal zero. It's just gonna be a big number, definetly not equal to zero.
So that means actually that his conjecture is not valid.
It's not true and what I just did over here: n equals 100, that's a counter example.
Because it's a specific example that shows that his conjecture is wrong.
I can plug in, for example, 100 into that expression for t and my answer is not zero. Therefore, he is wrong.
So a counter example is just one example to prove someone is wrong.
That counter prefix means like going against the clean.
Alright, let's look at example B:
*Arthur is making figures for a graphic art project. He drew polygons and some of their diagonals.*
And here we have four examples, based on these examples, Arthur made this conjecture:
*if a convex polygon has n sides, then there are n-2 triangles drawn from any given vertex of the polygon.*
So let's just think about what that means. He is saying that, if the shape has n sides,
for example, this would be n equals 3, three sides,
four sides,
five sides,
six sides.
He is saying that there are always n minus 2 triangles.
So, for example, if n is five, 5-2 is 3, he is saying, in this example, there are 3 triangles: one, two, three.
And in the next one that are four triangles, which is 6-2.
So, the question is: is Arthur's conjecture correct? Can you find a counter example to his conjecture?
Well, his conjecture certainly appears to be correct based on the four examples that he has done.
And we could do more examples and it would actually turn out that any example you'll try, it will be true,
but you still haven't proven it, if you've just looked at examples, because it is still possible
that that's another example out of there that you haven't thought out. That would be a counter example to his conjecture.
So what we should say is:
his conjecture appears to be true,
but it still needs to be proven.
Because just looking at examples isn't an official really proof for sure that it will always be true.