Nov 26, 2012
MATHEMATICAL PROOF 2_Docemented by Marsigit
Once a theorem is proved, it can be used as the basis to prove further statements. The so-called foundations of mathematics are those statements one cannot, or need not, prove. These were once the primary study of philosophers of mathematics. Today focus is more on practice, i.e. acceptable techniques.
Some common proof techniques are:
Direct proof: where the conclusion is established by logically combining the axioms, definitions and earlier theorems
Proof by induction: where a base case is proved, and an induction rule used to prove an (often infinite) series of other cases
Proof by contradiction: where it is shown that if some property were true, a logical contradiction occurs, hence the property must be false.
Proof by construction: constructing a concrete example with a property to show that something having that property exists.
Proof by exhaustion: where the conclusion is established by dividing it into a finite number of cases and proving each one separately