Proof by Deduction
Maths: Edexcel A Level Pure Maths, topic 1.3.2
What is proof by deduction?
Proof by deduction is a proof that consists of using known theorems to prove a given statement is always true.
How do you prove by deduction?
- Proof by deduction begins with a known theorem.
- We use a series of logical steps to go from the theorem to the final statement.
- You need to write out each step clearly and consider all potential cases.
- This statement should either prove the conjecture is always true or that it must be false.
Example
- Prove that the sum of two odd numbers is even.
Assumptions
- If and are odd, we can write them as:
- and
- Where and are integers.
Sum
- Adding and together, we get:
- Collecting terms in a sum like this is known to be always true, so it's a valid step in the proof.
Inspection
- In order for to be even, we need to be able to write it as , where is an integer.
- By inspecting the equation for we obtained earlier, we can rewrite it as:
- As and are integers, is also an integer.
- This means is an even number, for all possible odd values of and .
Final statement
- The sum of two odd numbers is an even number.
Full proof
Test yourself on Proof by Deduction
What are the steps involved in proof by deduction?