Let p(x) be a mathematical sentence involving a symbol x, so that, when x is a particular element of some universal set, p(x) is a statement that is either true or false. What may be of concern is the proving or disproving of the supposed theorem that p(x) is true for all x in the universal set. The supposed theorem can be shown to be false by producing just one particular element of the universal set to serve as x that makes p(x) false. The particular element produced is a counterexample. For example, let p(x) say that cos x+sin x=1, and consider the supposed theorem that cos x+sin x=1 for all real numbers x. This is demonstrably false (though p(x) may be true for some values of x) because x=π/4 is a counterexample: cos(π/4)+sin(π/4) ≠ 1.
Subjects: Mathematics — Psychology.