Core Ideas of Proof by Contradiction

The assumption creates a conditional: ¬P → (Q ∧ ¬Q). Since the consequent is a logical contradiction (always false), the antecedent (¬P) must be false by modus tollens, proving P is true

These can be proven directly, but contrapositive or even induction are also valid. Proof by contradiction is most natural for non-existence claims because assuming existence gives you a concrete object to manipulate. For example, to prove there is no greatest integer, we assume the opposite: that there is a greatest integer, say N. Then, consider N+1, which is an integer greater than N, contradicting the maximality of N. The other statements are typically proven directly through algebraic manipulation or definition.

The contradiction arises from the combination of the negation assumption and established axioms. Since a contradiction is always false, it indicates that the assumption must be incorrect. Therefore, we reject the assumption, which means the negation is false, and thus the original statement is true.

The original statement is an implication: if P → Q, where P is "3n+2 is odd" and Q is "n is odd". The negation of an implication is P ∧ ¬ Q. Therefore, we assume that 3n+2 is odd and n is even. If n is even, then 3n is even (since 3 times an even number is even), and 3n+2 is even (even plus even is even), contradicting the assumption that 3n+2 is odd.

The core of proof by contradiction is to assume the negation of the statement, not the statement itself. If one mistakenly assumes the original statement and derives a contradiction, that would disprove the original statement, which is the opposite of the intended goal. Always ensure the assumption is the negation.

A contradiction must be a statement that is false in all contexts. "0 = 1" is mathematically false, while the other options are true identities. Such a falsehood demonstrates that the assumption leads to an impossibility, fulfilling the requirement for a contradiction.

The proof follows a standard structure: assume √3 is rational, so √3 = a/b where a and b are integers with no common factors. Squaring both sides gives 3 = a²/b², so a² = 3b². This implies a² is divisible by 3, so a is divisible by 3 (since 3 is prime). Write a = 3k, then substitute to get 3b² = 9k², so b² = 3k², meaning b² is divisible by 3, so b is divisible by 3. This contradicts that a/b is in lowest terms, as both a and b share a factor of 3.

After assuming the negation and deriving a contradiction, the final step is to explicitly state that the original statement must be true. This conclusion follows logically from the rejection of the negation, completing the proof.

The statement to prove is that no integers satisfy the equation. The negation is that there exist integers a and b such that 2a + 4b = 1. Assuming this, we note that 2a + 4b = 2(a + 2b) is even, while 1 is odd, leading to the impossible conclusion that an even integer equals an odd integer. Thus, the assumption is false.

Proof by contradiction is classified as an indirect proof method because it does not directly prove the statement but instead shows that the negation is false. It is used in various fields, including logic and philosophy, and always involves an assumption. It is not universally the best method; its effectiveness depends on the context.

After assuming there are finitely many primes, we form N = (product of all primes) + 1. N is not divisible by any prime in the list (since division by any prime leaves remainder 1). Therefore, N must be prime or have a prime factor not in the list. In either case, there is a prime not in the list, contradicting the assumption that the list is complete.

The first step is to explicitly state: 'Assume, for contradiction, that ¬P holds.' This is the supposition that will be refuted by deriving logical consequences leading to a contradiction.

The irrationality of √5 is a classic example proven by contradiction, similar to √2. We assume √5 is rational, express it as a fraction in lowest terms, and show that both numerator and denominator are divisible by 5, contradicting the lowest terms assumption. The other properties are typically established through direct definitions or proofs.

After assuming ¬P, we apply inference rules (modus ponens, universal instantiation) and algebraic manipulations to sequentially derive intermediate implications from this assumption. These deductions are sequential steps that aim to reach a contradiction, such as a false statement or an inconsistency with established facts.

It relies on the Law of Excluded Middle: P∨¬P is a tautology. By showing ¬P leads to contradiction, we affirm P must hold, as there is no third truth value.