ExampleProof by Division into CasesThe following argument form is valid:It often happens t...

Example

Proof by Division into Cases

The following argument form is valid:

It often happens that you know one thing or another is true. If you can show that in either case a certain conclusion follows, then this conclusion must also be true. For instance, suppose you know that x is a particular nonzero real number. The trichotomy property of the real numbers says that any number is positive, negative, or zero. Thus (by elimination) you know that x is positive or x is negative. You can deduce that x2 > 0 by arguing as follows:

x is positive or x is negative.

If x is positive, then x2 > 0.

If x is negative, then x2> 0.

∴ x 2> 0.

The rules of valid inference are used constantly in problem solving. Here is an example from everyday life.