## Quick Reference

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.