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Counterexample in Math: Definition & Examples

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  • 0:00 What Is a Counterexample?
  • 0:34 Examples
  • 2:55 Lesson Summary
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Lesson Transcript
Instructor: Barry Rollins

Barry has taught mathematics at the college and high school level and has a master's degree in teaching secondary mathematics.

Counterexamples are a useful tool in mathematics. Learn what a counterexample is and how it can be used to prove the boundaries of theorems. You will also look at some examples across different branches of mathematics.

What Is a Counterexample?

A counterexample is a special kind of example that disproves a statement or proposition. Counterexamples are often used in math to prove the boundaries of possible theorems. In algebra, geometry, and other branches of mathematics, a theorem is a rule expressed by symbols or a formula. Counterexamples are helpful because they make it easier for mathematicians to quickly show that certain conjectures, or ideas, are false. This allows mathematicians to save time and focus their efforts on ideas to produce provable theorems.

Examples

Counterexamples are often used in math, but the truth is, counterexamples are all around us. Consider the statement, 'All mathematicians have crazy hair.' Now consider the difficulty you would encounter trying to prove this statement. Would this even be possible? It can often be very difficult to prove a proposition is true, so our efforts are better focused to find a counterexample and prove the proposition false. It only takes one counterexample to make the proposition false.

In geometry, the study of logic and proof is very important. Counterexamples are often discussed, as they can be used to disprove conditional statements. A conditional statement has two parts, such as: 'If you study algebra every night, then you will make 100 on your next test.' There are likely many examples that could be provided to make this statement true, but again, we only need one counterexample to prove it false. To find a counterexample to a conditional statement, you need an example to make the initial condition true, but at the same time, make the concluded statement false. Any student who studies algebra every night would satisfy the initial condition, and if one of these students did not make 100 on their next test, they would not satisfy the concluded statement. Thus, the first part is true, while the second is false. This student would be a counterexample.

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