# Proof by deduction

**Proof by deduction** is a process in maths where we show that a statement is true using well-known mathematical principles. With this in mind, try not to confuse it with Proof by Induction or Proof by Exhaustion.

The word **deduce** means to establish facts through reasoning or make conclusions about a particular instance by referring to a general rule or principle. Furthermore, *deduction* is the noun associated with the verb *deduce*. It follows that proof by deduction is the demonstration that something is true by showing that it must be true for all instances that could possibly be considered. Hence, it is not sufficient to check that a statement is true for a few example numbers – this is a mistake that is often made.

In maths, proof by deduction usually requires the use of algebraic symbols to represent certain numbers. For this reason, the following are very useful to know when trying to prove a statement by deduction:

- Use to represent any integer. Use and to represent any two integers.
- This means we should use , and to represent 3 consecutive integers. Alternatively, use , and or equivalent.
- In addition, if represents any integer, then represents any even integer and represents any odd integer.
- It follows that and represent any two consecutive even numbers; and represent any two consecutive odd numbers. Alternatively, we can use and to represent two consecutive even numbers, or and to represent two consecutive odd numbers.
- Furthermore, use and to represent any two square numbers.
- and represent any two consecutive square numbers and so on…

Note that a certain amount of algebra is required when completing proofs. For example, expanding as .

## Examples

## Videos

How to use Proof by Deduction to show that the given expression is always even, regardless of the choice of integer to put in.

Sample Assessment question requiring the use of Proof by Deduction to show that an algebraic inequality holds followed by a Disproof by Counterexample question.

Proof of the Quadratic Formula using Proof by Deduction

By recognising a hidden quadratic, Proof by Deduction can be used to show tricky inequalities.

## Extra Resources

Take the mathematical proof test – a selection of exam style questions on Proof by Deduction, Proof by Exhaustion and Disproof by Counterexample.