Methods of proving

You can prove mathematical statements on many ways. In the book tilted Plane and Solid Geometry by Miss Julie Amado Buasen, she enumerated seven methods in proving mathematical statements.      

1. Direct Proof  is one of the most familiar forms of proof. We use it to prove statements of the form “if p then q” or “implies q” which we can write as p ⇒ q. The method of the proof is to takes an original statement p, which we assume to be true, and use it to show directly that another statement q is true. Therefore, a direct proof has the following steps:

·         Assume the statement p is true.

·         Use what we know about p and other facts as necessary to deduce that another statement q is true, that is show p ⇒ q is true.

Example: Prove that if a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

Proof: Let any straight line AB standing on the straight line CD make the angles CBA and ABD. I say that either the angles CBA and ABD are two right angles or their sum equals two right angles.    

                

                 Now, if the angle CBA equals the angle ABD, then they are two right angles by proposition 11. However, if not, draw BE from the point B at right angles to CD. Therefore, the angles CBE and EBD are two right angles by common notion 2.

                 Since the angle CBE equals the sum of the two angles CBA and ABE, add the angle EBD to each, therefore the sum of the angles CBE and EBD equals the sum of the three angles CBA, ABE, and EBD. Again, since the angle DBA equals the sum of the two angles DBE and EBA, add the angle ABC to each, therefore the sum of the angles DBA and ABC equals the sum of the three angles DBE, EBA, and ABC as proposed by common notion 1.

                 However, the sum of the angles CBE and EBD was also proved equal to the sum of the same three angles, and things, which equal the same thing, also equal one another, therefore the sum of the angles CBE and EBD equals the sum of the angles DBA and ABC. However, the angles CBE and EBD are two right angles; therefore, the sum of the angles DBA and ABC also equals two right angles.

                 Therefore, if a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

Simply speaking, the conclusion was established by logically combining the axioms, definitions and earlier theorems. This method of proving was set to be useful in the olden times when things are being discovered. There was no proper proving and this was considered the acceptable one.

2. Contradiction is also known as reduction od absurdum, Latin for “by reduction by the absurd”. It shows that if some statements are so, a logical contradiction occurs, hence the statements must be not so. This method is perhaps the most prevalent of mathematical proofs.

          

3.  Construction or also known as proof by example, is the construction of a concrete example with a  property to show that something having that property exists. Take for the example given in the direct proof. We can prove its existence by constructing an illustration of the proposition. 

4. Exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large.

5    5.  Visual Proof. Although not a formal proof, a visual demonstration of a mathematical  theorem is sometimes called "proof without words".

^   6. Mathematical Induction. In proof by mathematical induction, first a base case is proved then an induction rule is used to prove series of other cases.

             Mathematicians often used the term proof by induction. However, the term proof by induction may also be used in logic to mean an argument that uses inductive reasoning.

     7. Two-column proof. A particular form of proof using two parallel columns is often used in elementary geometry classes. the proof is written as a series of lines in two columns. in each line, the left hand column contains a proposition  while the right hand column contains brief explanation of how corresponding proposition in the left hand column is either an axiom, hypothesis or  can be logically derived form previous propositions.