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Contradiction and Counterexamples
So far, we have learned to prove that statements are true. In this lesson, we learn strategies to prove that statements are not true.
So far, we have learned to prove that statements are true. In this lesson, we learn strategies to prove that statements are not true.
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A rational number is one that can be written in the form
Note that any fraction can be reduced to the point where the numerator and denominator share no common factors:
To prove a statement by contradiction, we essentially show that if the statement were not true, math would "break".
In practice, we assume the opposite of the statement, and and derive a contradiction - a mathematical inconsistency.
A prime number is a natural number greater than 1 that has exactly two distinct positive divisors: 1 and itself. For example, the set of prime numbers begins as
Notice that 1 is not included in this list because it has only one positive divisor (namely, itself), and thus does not meet the definition.
The easiest way to show a statement is not true is to find a specific example for which it is not true.