Theorem and Converse

After axioms come the claims you must work for. A conjecture is a guess that still needs proof. A theorem is a statement that can be proved from axioms, postulates, definitions, and earlier theorems. A corollary is a theorem that follows so quickly from another that it needs almost no extra argument. The converse of a statement interchanges the given and the to-prove. The converse need not be true, so it needs its own check.

1. CONJECTURE TO THEOREM: A conjecture still needs proof: 'I think every even number greater than 2 is a sum of two primes' is a famous guess, not a Class 9 theorem. A theorem can be proved from axioms and earlier results, and then you may quote it. A corollary follows quickly: if you have proved the angle sum of a triangle is 180 deg, then each angle of an equilateral triangle is 60 deg. 2. THE CONVERSE: To write the converse, interchange the given and the to-prove. 'If a triangle is equilateral, then it is equiangular' has converse 'if a triangle is equiangular, then it is equilateral'. That pair happens to be true, but many converses fail. 'If a figure is a square, then it is a rectangle' is true; the converse 'if a figure is a rectangle, then it is a square' is false.

Exam papers love the word converse because students copy the original proof and think they have proved the swap. You have not. Treat the converse as a new proposition: new Given, new To prove. If a counter-example exists, stop and say the converse is false.

A conjecture is not a theorem until it is proved. A corollary is a fast follow-on, not a new deep proof. The converse swaps given and to-prove and need not be true. Never assume the converse of a theorem without checking.