This is a free lesson from our course in Algebra I
In this lesson, you'll learn how to use the counter example to disprove a
conjecture. A conjecture is a mathematical statement which has been proposed as
a true statement, but which no one has yet been able to prove or disprove. Once
the conjucture is proven, it is known as a theorem and and joins the mathematical
facts. The counterexample, like mentioned above, can be used to disprove a mathematical
conjecture.For example:
•
all prime numbers are odd (conjecture)
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but 2 is prime number (counter example)
•
the counter example above disproves the conjecture, hence we can conclude that not all prime numbers are odd.
(More text below video...)
(Continued from above) Let's look at one more example:
Prove that “For every positive integer n, n! <= n^{2}.”
Start testing some cases say, n = 1, 2, 3 etc.
It might seem like it is true for some cases but how far do you test, say n = 4.
Here, you get n! = 24 and n^{2} = 16 which is a counter example for this theorem. Hence,even finding a single case that doesn’t satisfy the condition is enough to disprove the theorem.
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