Proof by Contradiction

Definition

Proof by contradiction is a method of mathematical proof in which the assumption that a statement is false leads to a contradiction, thereby proving that the statement must be true. This technique is particularly useful when direct proofs are challenging, allowing mathematicians to establish the truth of a proposition by showing that assuming its negation results in an impossible situation. It often involves using existing theorems or definitions to derive consequences that contradict known facts.

5 Must Know Facts For Your Next Test

1. Proof by contradiction often starts by assuming that the statement to be proven is false, then logically deducing consequences from this assumption.
2. If a contradiction arises during this deduction process, it indicates that the initial assumption must be incorrect, thus proving the original statement true.
3. This method can be particularly effective in proving the irrationality of numbers, such as showing that √2 is not a rational number.
4. In the context of sequences, proof by contradiction can demonstrate properties like the convergence of monotone sequences by showing what happens if they were assumed to diverge.
5. Mathematicians often use proof by contradiction when dealing with limits and infinities, where direct proofs may be less intuitive.

Review Questions

• How does proof by contradiction differ from direct proof in establishing the properties of sequences?
  • Proof by contradiction differs from direct proof in that it starts with the assumption that the conclusion is false and seeks to find a logical inconsistency. In establishing properties of sequences, proof by contradiction may reveal contradictions in limit behaviors or sequence convergence, while direct proof builds a chain of reasoning directly linking premises to the conclusion without assuming its negation.
• In what ways can proof by contradiction demonstrate the convergence of monotone sequences?
  • Proof by contradiction can demonstrate the convergence of monotone sequences by assuming that such a sequence does not converge and then deriving contradictions based on its behavior. For instance, one might assume a bounded monotone increasing sequence diverges, leading to a conclusion that contradicts its boundedness. This process confirms that if the conditions hold, then the sequence must converge.
• Evaluate the importance of proof by contradiction in establishing foundational results in mathematical analysis, particularly regarding Cauchy sequences.
  • Proof by contradiction plays a crucial role in establishing foundational results in mathematical analysis, especially concerning Cauchy sequences. By employing this method, mathematicians can prove significant properties like completeness and convergence without needing direct constructions. For example, proving that every Cauchy sequence converges in complete spaces can rely on contradictions arising from assuming divergence. Such foundational proofs underpin much of real analysis and ensure rigorous understanding of limits and continuity.

Related terms

Contrapositive: The contrapositive of a statement reverses and negates both the hypothesis and conclusion, which is logically equivalent to the original statement.

Direct Proof: A direct proof establishes the truth of a statement by a straightforward chain of logical deductions from accepted premises or previously established results.

Cauchy Sequence: A sequence is called a Cauchy sequence if, for every positive real number, there exists an index such that the distance between any two terms after that index is less than that number.