Counterexample Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

medium
Disprove: 'For all real numbers xx, x2>xx^2 > x.'

Solution

  1. 1
    Try x=0x = 0: 02=00^2 = 0 and 0>ฬธ00 \not> 0. This is a counterexample.
  2. 2
    Alternatively, try x=12x = \frac{1}{2}: (12)2=14<12(\frac{1}{2})^2 = \frac{1}{4} < \frac{1}{2}, another counterexample.
  3. 3
    Either one suffices to disprove the statement.

Answer

Counterexample:ย x=0,ย sinceย 02=0>ฬธ0.\text{Counterexample: } x = 0, \text{ since } 0^2 = 0 \not> 0.
When disproving inequalities, test boundary values (like 0, 1, or fractions) as they often reveal counterexamples.

About Counterexample

A counterexample is a specific instance that satisfies the hypothesis of a claim but contradicts its conclusion, thereby disproving the general statement.

Learn more about Counterexample โ†’

More Counterexample Examples

Example 1 easy

Disprove: 'All prime numbers are odd.'

Example 3 easy

Find a counterexample to disprove: 'The sum of any two prime numbers is even.'

Example 4 easy

Find a counterexample to disprove: 'If an integer is divisible by 6, then it is divisible by 4.'

โ† All Counterexample Examples Counterexample Concept