Counterexample - definition and examples

A counterexample is any example proving that a statement is false. You only need one counterexample to prove that a statement is false.  

Using counterexamples

Is each statement true or false? If the statement is false, give a counterexample.


The square root of a number is always smaller than the number.

The statement is false. The square root of 0.25 is a counterexample. 

√(0.25) = 0.5 and 0.5 is not smaller than 0.25


If a number is an irrational number, then squaring the number will also give an irrational number.

The statement is false. The √(3) is a counterexample. 

√(3) = 1.73205080757 is irrational

However, [√(3)]2 = √(3) × √(3) = √(3×3) = √(9) = 3 and 3 is a rational number


All birds can fly

The statement is false. Penguin is a counterexample. 


All integers are rational numbers

The statement is true


No negative number is less than its absolute.

The statement is false. -1 is a counterexample.

|-1| = 1 and -1 is less than 1


If an animal lives on land, then it is a mammal.

The statement is false. A whale is a counterexample.

A whale is an aquatic mammal.

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