What Is a Counterexample in Algebra?

Writing a counterexample is not as difficult as you imagine.
••• Jupiterimages, Brand X Pictures/Brand X Pictures/Getty Images

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.

Famous Counterexamples

"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.

Related Articles

What Is the Meaning of Unbounded & Bounded in Math?
Advantages & Disadvantages of Boolean Logic
Definition of Successor and Predecessor in Math
How to Determine If Matrices Are Singular or Nonsingular
The Four Types of Multiplication Properties
How to Find Consecutive Integers
How to Solve Inequalities
How to Find the Domain of a Set of Numbers
How to Solve Linear Systems Algebraically
What Makes a Relation a Function?
How to Find the Slope of a Nonlinear Line
Advantages & Disadvantages of Boolean Logic
Differences Between Absolute Value & Linear Equations
How to Find the Inverse of a Function
What Is the Meaning of Quantitative Observation?
How to Explain Different Types of Proofs in Geometry
What Is a Regression Line?
How to Find Quadratic Equations From a Table
Ways to Tell if Something Is a Function