In mathematics, a counterexample is used to disprove a statement. If you want to prove that a statement is true, you must write a proof to demonstrate that it is always true; giving an example is not sufficient. Compared to writing a proof, writing a counterexample is much simpler; if you want to show that a statement is not true, you only need to provide one example of a scenario in which the statement is false. Most counterexamples in algebra involve numeric manipulations.
Two Classes of Mathematics
Proof-writing and finding counterexamples are two of the primary classes of mathematics. Most mathematicians focus on proof-writing to develop new theorems and properties. When statements or conjectures cannot be proved true, mathematicians disprove them by giving counterexamples.
Counterexamples Are Concrete
Instead of using variables and abstract notations, you can use numeric examples to disprove an argument. In algebra, most counterexamples involve manipulation using different positive and negative or odd and even numbers, extreme cases and special numbers like 0 and 1.
One Counterexample Is Sufficient
The philosophy of the counterexample is that if in one scenario the statement does not hold true, then the statement is false. A non-math example is "Tom has never told a lie." To show this statement is true, you have to provide "proof" that Tom has never told a lie by tracking every statement Tom has ever made. However, to disprove this statement, you only need to show one lie that Tom has ever spoken.
"All prime numbers are odd." Although almost all prime numbers, including all primes above 3, are odd, "2" is a prime number that is even; this statement is false; "2" is the relevant counterexample.
"Subtraction is commutative." Both addition and multiplication are commutative -- they can be performed in any order. That is, for any real numbers a and b, a + b= b + a and a * b = b * a. However, subtraction is not commutative; a counterexample proving this is : 3 - 5 does not equal 5 - 3.
"Every continuous function is differentiable." The absolute function |x| is continuous for all positive and negative numbers; but it is not differentiable at x = 0; since |x| is a continuous function, this counterexample proves that not every continuous function is differentiable.
- "Counterexamples in Analysis"; Bernard Gelbaum, John Olmsted; 1992; pg.v-vi, 38;
About the Author
Alice Lou holds a B.S. in mathematics from Columbia College and an M.S. in operations research from Columbia University. She is also a chartered financial analyst level II candidate. Lou currently teaches undergraduate mathematics at a four-year college.
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