Proof by contradiction

Definition

Proof by contradiction is a mathematical technique where the validity of a statement is established by assuming the opposite of what is to be proven, leading to a contradiction. This method effectively shows that if the opposite were true, it would create an impossible scenario, thereby confirming that the original statement must be true. It's commonly used in economics to demonstrate the validity of theories or propositions by logically ruling out alternatives.

5 Must Know Facts For Your Next Test

1. In proof by contradiction, you start by assuming the negation of the statement you want to prove, and then derive a conclusion that contradicts known facts or assumptions.
2. This technique is powerful in mathematics and economics because it allows for the exploration of complex arguments by narrowing down possibilities.
3. Common examples of proof by contradiction include demonstrating properties of numbers, such as irrationality or properties of functions.
4. It can often simplify arguments because showing that all alternatives lead to contradictions makes the original statement more compelling.
5. Proof by contradiction is essential for establishing theoretical results in economics, especially when analyzing market behaviors or proving optimal conditions.

Review Questions

• How does proof by contradiction help clarify complex economic theories?
  • Proof by contradiction clarifies complex economic theories by allowing economists to examine various assumptions and demonstrate their limitations. By assuming the opposite of a theory and showing that it leads to contradictions with known principles or data, economists can strengthen their argument for the original theory. This method provides a clear path to validating hypotheses that may otherwise be difficult to prove directly.
• Discuss how proof by contradiction can be applied to demonstrate a fundamental economic principle, such as market equilibrium.
  • To demonstrate market equilibrium using proof by contradiction, one might assume that a market is not in equilibrium, meaning excess supply or demand exists. This assumption would lead to contradictions with the idea that prices adjust until supply equals demand. If prices are not allowed to change under this assumption, it contradicts the basic market behavior observed in real-world scenarios. Hence, proving that any deviation from equilibrium leads to contradictions reinforces that markets do tend toward equilibrium.
• Evaluate the role of proof by contradiction in advancing mathematical economics and its implications for real-world applications.
  • Proof by contradiction plays a crucial role in advancing mathematical economics by providing robust methods for proving critical concepts and relationships. By establishing theoretical foundations through this technique, economists can derive results that have real-world implications, such as optimal pricing strategies and market behavior predictions. Understanding how these proofs work enhances our grasp of economic models and prepares us for applying these theories effectively in policy-making and economic forecasting.

Related terms

Contrapositive: The contrapositive of a statement is formed by negating both the hypothesis and the conclusion and reversing their positions, which can be used to prove implications.

Direct proof: A method of proof that establishes the truth of a statement by a straightforward chain of logical deductions from accepted premises.

Logical equivalence: Two statements are logically equivalent if they imply each other, meaning if one is true, the other must also be true.