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: If
then
This statement is 'obviously true', but if
and
then
but
so the statement is not true.
If
and
are both greater than or equal to zero then the statement is true.
Formally, ifand
then
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.
Takeand
then
but
Example: Ifandare different irrational numbers, then
and
are both irrational.
Take
and
so that
and
so thatandare both rational.
Example: Ifandare different irrational numbers, then
and
are both irrational.
Take
and
so that
and
so thatandare both rational.