Understanding Proof by Contradiction and Contrapositive in Mathematics

The principles of proof by contradiction and the law of contrapositive are fundamental tools in mathematical logic and reasoning. These techniques help mathematicians and logicians to prove or disprove statements. This article will explore the methods, illustrate examples, and delve into the underlying logic behind both techniques.

Proof by Contradiction

Proof by contradiction is a method of proving a statement by assuming the opposite of what you want to prove and then showing that this assumption leads to a contradiction. According to Fermat's Last Theorem, for any integer ( n > 2 ), there are no positive integers ( a, b, ) and ( c ) such that ( a^n b^n c^n ).

Historical Example: Fermat's Last Theorem (n4)

Fermat used the method of infinite descent to prove his last theorem for ( n 4 ). The method of infinite descent involves assuming there exists a positive integer solution ( x ) which is the lowest possible value. Through mathematical manipulation and calculations, it is shown that this assumption leads to an even smaller positive integer solution, which contradicts the original assumption that ( x ) is the smallest. This contradiction means the initial assumption must be incorrect, proving that no such ( x ) exists.

Much later, Andrew Wiles, drawing upon the work of many mathematicians including Kenneth Ribet, provided a more sophisticated proof using the modular forms and elliptic curves. He proved that if the last theorem was false, there would exist a semistable elliptic curve that is not a modular form. The Taniyama-Shimura conjecture suggests that all semistable elliptic curves are modular forms. Wiles' proof of the conjecture effectively proved Fermat's Last Theorem.

The Law of Contrapositive

The law of contrapositive states that if the conditional statement "if A is true then B is true" is true, then the contrapositive "if not B then not A" is also true. The law of contrapositive is a logical equivalence, meaning that the two statements are logically equivalent.

Examples of Contrapositive Statements

Consider the statement "if ( x > 0 ) then ( x^2 > 0 )". The contrapositive of this statement is "if ( x^2 leq 0 ) then ( x leq 0 )". Both statements are logically equivalent and can be proven using the same core principles.

Types of Statements

Mathematical statements can be classified into different types under the Aristotelian square of opposition:

A claim: ALL s ARE p O claim: SOME s ARE NOT p E claim: ALL s ARE NOT p I claim: SOME s ARE p

The contrapositive is only valid for A claims and O claims. For A claims, "ALL s ARE p" can be written as "if s then p". The contrapositive of "if s then p" is "if not p, then not s". Similarly, for O claims, "SOME s ARE NOT p" can be written as "if s, then not p", and its contrapositive is "if p, then not s".

Modus Ponens and Modus Tollens

The proof by contradiction often relies on the application of modus ponens and modus tollens.

Modus Ponens: IF s THEN p, and s, therefore p Modus Tollens: IF s THEN p, and not p, therefore not s

The contrapositive essentially uses the modus tollens argument structure. In the example of "IF s THEN p", the contrapositive "IF not p THEN not s" is derived by negating both the consequent and antecedent and switching their positions.

In conclusion, proof by contradiction and the law of contrapositive are powerful tools in mathematics. They help to establish the truth of statements by considering the logical implications of their negations. Understanding these concepts is crucial for anyone delving into advanced mathematics or rigorous logical reasoning.