Basics of Proof

Proof in mathematics is a way of showing that a principle, rule or formula is universally true.
At foundation level, you’ll mostly deal with reasoning using a set of given rules or from known truths.
This might include showing that a certain calculation leads to a specific result or that a formula holds in all cases examined.

Types of Proof

There are several different types of proof that you may explore in this course. These include direct proof, proof by example, proof by contradiction, and proof by induction.
Direct proof proceeds from given facts via a series of logical steps to the conclusion.
Proof by example involves showing that a rule holds for a specific case.
Proof by contradiction involves supposing the opposite of what you want to prove and showing this leads to a contradiction.
Proof by induction is a method based on proving a base case, then proving that if any one case is true the next must be true.

Geometry is an area where proof is often used. For example, you may prove that the angles in a triangle add up to 180 degrees.
Algebra also requires understanding of how to prove identities, demonstrate properties of numbers, or establish validity of formulas.

A common mistake is to assume what you’re trying to prove. This is known as begging the question and is not a valid form of proof.
Failing to make sure your logic applies in all relevant cases can lead to incorrect conclusions.
It’s important to understand the difference between providing an example and proving a rule. An example might illustrate a concept, but it doesn’t prove it without exception.