Proof by contradiction

Definition

Proof by contradiction is a method of mathematical proof that establishes the truth of a statement by assuming the opposite is true and showing that this assumption leads to a contradiction. This technique is powerful because it often simplifies the reasoning process by allowing the use of logical deductions to arrive at an impossible conclusion, thereby affirming the original statement. This method connects closely with rules of inference and validity, as it relies on sound logical principles to derive conclusions.

5 Must Know Facts For Your Next Test

1. Proof by contradiction relies on the principle of the excluded middle, which states that for any proposition, either it is true or its negation is true.
2. This method is particularly useful in proving statements about irrational numbers or properties of integers, where direct proof might be cumbersome.
3. When using proof by contradiction, if you assume the negation of the statement you're trying to prove and reach a logical inconsistency, you validate the original statement.
4. It's essential to clearly identify what assumption leads to the contradiction so that the proof remains logically sound.
5. Proof by contradiction is often used in conjunction with other proof methods, enhancing its effectiveness in more complex arguments.

Review Questions

• How does proof by contradiction utilize logical principles to establish the truth of a statement?
  • Proof by contradiction utilizes the principle of the excluded middle, which asserts that every proposition must be either true or false. By assuming that the statement we want to prove is false, we derive logical consequences from this assumption. If these consequences lead to an impossibility or contradiction, it confirms that our initial assumption must be incorrect, thereby proving that the original statement is true.
• Compare proof by contradiction with direct proof in terms of their approaches to validating mathematical statements.
  • Proof by contradiction and direct proof are both methods used to validate mathematical statements, but they take different approaches. Direct proof starts with established premises and uses logical deductions to arrive directly at the conclusion. In contrast, proof by contradiction assumes the opposite of what you want to prove and demonstrates that this assumption leads to a logical inconsistency. While direct proofs can be more straightforward, proof by contradiction can sometimes handle more complex scenarios where direct methods struggle.
• Evaluate how effective proof by contradiction is in addressing statements regarding irrational numbers compared to other types of proofs.
  • Proof by contradiction is particularly effective for proving statements about irrational numbers, such as showing that √2 is irrational. When employing this method, one assumes that √2 can be expressed as a fraction (the opposite of what we're trying to prove) and derives consequences based on this assumption. This approach leads to contradictions involving integer properties that cannot hold true, confirming that √2 cannot be expressed as a fraction. In cases like this, proof by contradiction often provides a clearer path than direct proofs, showcasing its strength in addressing such statements.

Related terms

Contrapositive:

A contrapositive is a logically equivalent statement formed by negating both the hypothesis and conclusion of a conditional statement and switching them.

Logical Negation: Logical negation is an operation that takes a proposition and flips its truth value, turning a true statement into a false one and vice versa.

Direct Proof:

A direct proof is a straightforward method of proving a statement by assuming the premises are true and logically deriving the conclusion from them.