# Proof by Exhaustion

**Proof by Exhaustion**Â is the proof that something is true by showing that it is true for each and everyÂ case that could possibly be considered. This is also known asÂ **Proof by CasesÂ **– see Example 1 in the Proof by Deduction Examples lesson.Â This is different from Proof by Deduction where we use algebraic symbols and construct logical arguments from known facts to show that something is true for all numbers. For the case of Proof by Exhaustion, we show that a statement is true for each number in consideration (or subsets of numbers). Proof by Exhaustion also includes proof where numbers are split into a set of exhaustive categories and the statement is shown to be true for each category. Proof by Deduction can then be used within the categories â€“ see Example 2 in the Proof by Deduction Examples lesson. In addition to the rules listed in the Proof by Deduction Notes, it is worth memorising the following sets of numbers:

- is known as the set of natural numbers. It is the set of all positive integers.
- is the set of all integers including those that are negative and 0. Note that the positive part is sometimes denoted which, of course, is simply .
- is the set of all rational numbers – these are numbers that can be represented as fractions and includes recurring decimals.
- is the set of all real numbers. This includes irrational numbers such as or but does not include the square root of negative numbers (these are known as imaginary numbers).

## Examples

## Videos

A detailed Proof by Exhaustion example showing that all square numbers are either a multiple of 4 or one more than a multiple of 4.

Another example of Proof by Exhaustion, slightly more involved than the previous video, showing that all square numbers are either a multiple of 3 or one more than a multiple of 3.