PROOFS IN MATHEMATICS

WHAT IS A PROOF ?


Proof of a mathematical statement consists of sequence of statements, each statement being justified with a definition or an axiom or a preposition that is previously established by the method of deduction using only the allowed logical rules.

Thus each proof is a chain of deductive arguments each of which has its premises and conclusions. Many a times, we prove a preposition directly from what is given in the preposition. But some times it is easier to prove an equivalent preposition rather than proving the preposition itself.

This leads to, two ways of proving a preposition directly or indirectly and the proofs obtained are called direct proofs and indirect proof and further each has three different ways of proving which is discussed below.


Direct proof

• Straight forward proof: It is a chain of arguments which leads directly from what is given or assumed, with the help of axioms, definitions or already proved theorems, to what is to be proved using rules of logic.
• Mathematical induction: It is a strategy of proving a preposition which is deductive in nature.
• Proof by cases or by exhaustion

Indirect proof 

• Proof by contradiction
• Proof by using contrapositive statement of the given statement
• Proof by counter example

Proofs are to Mathematics what calligraphy is to poetry. Mathematical works do consist of proofs just as poems do consist of characters.

—  VLADIMIR ARNOLD

Papyrus Oxyrhynchus 29 ( P. Oxy. 29 ) is a fragment of the second book of the elements of Euclid in Greek. It was discovered by Grenfell and Hunt in 1897 in Oxyrhynchus. One of the oldest surviving fragments of Euclid's elements, a textbook used for millennia to teach proof writing techniques.