Counterexamples
In inductive reasoning, we observe a variety of special cases and then find a common pattern. We then generalize this pattern as a common principle for all of the cases that are not specified.
| There are some patterns in which the generalization is valid for many cases, but one or more cases just doesn't follow that same pattern. These isolated cases are sufficient to reject the conjecture. Such cases are called counterexamples, like the penguin example from earlier in this section. |
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A Historical Example Recall that a prime number is a number that is divisible only by 1 and itself. 2, 3, 5, 7, 11, and 13 are examples of prime numbers. For many years mathematicians thought that any number in the form 2n − 1 is a prime number, where n itself is a prime. That is, replacing n by a prime number results in a number that is NOT divisible by any number except itself and 1. This expression could produce a large set of prime numbers, but not all. Their conjecture was based on the first series of prime numbers and was valid for such numbers. For example:
These all are prime numbers. This expression is valid for many other prime values of n. But, in the 17th Century, mathematicians verified that for n = 11, the result of This is not a prime number. It is divisible by 23, because 2047 = 23 × 89. As a result, the conjecture, which was used for a long time, lost its validity. No longer is it considered to be a universal rule that is true for all prime numbers. |
