## Proofs By ContradictionThis is also called
G.H. Hardy, A Mathematician's Apology, 1940. To prove a theorem , assume that the theorem does not hold. I.e, and prove that a contradiction (or absurditity results). Consider a simple example.
The original statement is The negation of this statement is Proof by Contradiction (Example)
Let us assume that the original statement is false. It's negation must be true for some . Therefore, there is a such that , is prime and is even, all at the same time. Since is even, we can write for some . Now, since , as well, cannot be equal to . As a result, can be written as the product of and another number that is not . Therefore, is composite. However, we assumed that was prime. A number cannot be prime and composite at the same time. Therefore, we have a contradiction.
Assuming the negation of the theorem leads to a contradiction. Therefore, the theorem is true. (QED) ## Proof By ContradictionLet us look at how such proofs look like. ## Universal Statement
Proof
Assume, for the sake of contradiction, that the statement does not hold. In other words, there is a number such that . We then use this to prove by contradiction. ## Universal ImplicationThis is a special case of universal statements.
Proof
Assume, for the sake of contradiction, that the statement does not hold. Therefore, there is a number such that holds but holds (or does not hold). Starting from this, we derive a contradiction. ## Existential StatementTo prove , assume that and derive the contradiction. ## Examples of Proof By ContradictionHere are some famous
Proof
Proof is by contradiction. Let us assume that there are finitely many (let us say ) primes. Therefore, the prime numbers are and every other number (except ) is composite. Consider the number . We know that it is composite since the number of primes is finite and . We showed using strong induction that `every number is divisible by some prime number`.Therefore, is divisible by one of . However, dividing by leaves a remainder of for each . As a result, we conclude that is a prime too.
This means that cannot be all the primes. Therefore, we have a contradiction.
Here is another one from Euclid.
Proof
Proof is again by contradiction. Let us assume that
for integers and . We also assume that
are at their We know that . Therefore, . This means that is even. Therefore, is even. However, cannot be even since would then not be in their lowest terms. Therefore, is odd. Since is even, we can write . Therefore . We conclude that . In other words, is even and hence is even.
This is a contradiction since cannot be odd and even at the same time. QED. |