Disproving a statement is often simpler than a proof. Finding a counterexample will often suffice. Since a statement cannot be both true and false, if you find a counterexample, the statement must be false.

Example: Ifthen

This statement is 'obviously true', but ifandthenbutso the statement is not true.

Ifandare both greater than or equal to zero then the statement is true.

Formally, ifandthen

Example: Ifthen

Again the statement is true if only positive numbers are considered – it is also true if only negative numbers are considered. If one number is negative and the other is positive then the statement is false.

Takeandthenbut

Example: Ifandare different irrational numbers, thenandare both irrational.

Takeandso thatand so thatandare both rational.

Example: Ifandare different irrational numbers, thenandare both irrational.

Takeandso thatand so thatandare both rational.