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
Prove that the difference between the squares of any two consecutive integers is equal to the sum of those integers.
Firstly, choose and to be any two consecutive integers. Next, take the squares of these integers to get and where 
. The difference between these numbers is 
. Adding together the original two consecutive numbers also gives 
. Hence, we have proved by deduction that the difference between the squares of any two consecutive integers is equal to the sum of those integers.
Prove that 
is always positive.
By completing the square can be written as 
. Note that 
is positive for any 
as it is a square number – adding 5 will retain its positivity.
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.