How to Prove Conjectures

TenMarks teaches you all about how to proving conjectures.

How to Prove Conjectures In this lesson, let’s learn how to prove conjectures. We will prove if conjectures are true or false. And all we have to do to determine if the conjectures are true or false is look for examples that prove if a conjecture is false. Such an example is called a Counterexample. If I can find even one example that does not hold for a conjecture, that’s called a counterexample that proves that the conjecture is not always true. So for example, let’s do the first one. It says points X, Y and Z are coplanar. Points are supposed to be coplanar if they lie on the same plane, right? So, there are two ways three points can occur. So, point way one is all three lie on the same line. They are collinear which means there’s a single line that connect through them. If there is a single line that can connect through them, I can always draw a plane that is along the same line as long as this lies on the plane so, I can always make a plane that makes these three points coplanar. The second is these three points may not be on a same line in which case I can draw a triangle through them and I can form a plane that holds this triangle. So, both of these cases, they always would have to be coplanar because whether they are in a straight line or they are not, I can always form a plane that holds all three points. So, this conjecture is true. I cannot come up with a counterexample. I look at every possible example. Let’s try the second problem. It says if n is an integer, then -n is positive. So, let’s do some examples. If n=-2, what is -n? -n is positive 2. So, this conjecture is also true. If n=-1 then -n is positive one, still also true. If n=3, -n is -3. And -3 is not positive, so I came up with an example where n is an integer where -n was not positive. This is the counterexample that proves that this conjecture is not always true. So, all we have to do to prove that a conjecture is not true is to show one counterexample as I did here.