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Contradiction and Counterexamples
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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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Rational Numbers
AHL AA 1.15
A rational number is one that can be written in the form
p=ba,a,b∈Z
Note that any fraction can be reduced to the point where the numerator and denominator share no common factors:
155=31
Proof by contradiction
AHL AA 1.15
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.
Prime Numbers
AHL AA 1.15
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
{2,3,5,7,11…}
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.
Counterexamples
AHL AA 1.15
The easiest way to show a statement is not true is to find a specific example for which it is not true.
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