A counter example is a set of conditions that makes a mathematical claim false.
Claim: If \(x\) is a real number, then \(x^2 > x.\)
Counter Example: The claim is not true because it is not a tautology. One counter example is \(x = 0.5.\) Then \(x^2 = 0.25 < 0.5 = x.\) So, \(x = 0.5\) falsifies the claim.
It is important to note that you only need one counter example to completely falsify a claim. A mathematical claim must be tautologically true, or in other words, a mathematical statement is only true if it is true for all values for which it makes a claim.
The claim "If \(x\) is a real number, then \(x^2 > x\)" is not false for all values of \(x,\) so it could be revised to a true statement.
Revised Claim: If \(x\) is a real number such that either \(x < 0\) or \(x > 1\), then \(x^2 > x.\)