Involves proving a statement true by reducing it to its opposite and demonstrating the absurdity of the opposite result

General
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involves proving a statement true by reducing it to its opposite and demonstrating the absurdity of the opposite result.

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What does “proving a statement true by reducing it to its opposite and demonstrating the absurdity of the opposite result” mean?

Answer:
This description refers to a mathematical and logical proof technique known as proof by contradiction. Here’s a detailed explanation of this method, articulated in a manner that is informative and easy to understand.

1. Definition of Proof by Contradiction

Proof by contradiction, also known as indirect proof, is a method to establish the validity of a statement by demonstrating that assuming the opposite leads to a contradiction. More formally, to prove a statement P is true, you assume that P is false (i.e., \neg P is true). You then show that this assumption leads to a logical inconsistency or an absurd result, thereby concluding that P must indeed be true.

2. Step-by-Step Explanation

  1. Assume the Negation:

    • Assume that the statement you want to prove, P, is false. This is denoted as \neg P.

  2. Derive Implications:

    • From the assumption \neg P, logically derive consequences and implications. Use axioms, definitions, and previously established results within your logical framework to investigate what the assumption \neg P entails.

  3. Find a Contradiction:

    • Reach a point where one of the derived implications contradicts a known fact (i.e., a theorem, axiom, or given condition). This contradiction demonstrates that the assumption \neg P cannot be true.

  4. Conclude the Proof:

    • Since assuming \neg P leads to a contradiction, it follows that \neg P must be false. Hence, P (the original statement) is true.

3. Example of Proof by Contradiction

To solidify understanding, consider a classic example: proving that \sqrt{2} is irrational.

Statement: \sqrt{2} is irrational.

Step-by-Step Proof:

  1. Assume Negation:

    • Assume \sqrt{2} is rational. Therefore, it can be expressed as a fraction \frac{a}{b} in its simplest form, where a and b are coprime integers (i.e., having no common factors other than 1).

  2. Square Both Sides:

    • If \sqrt{2} = \frac{a}{b}, then:
      2 = \left( \frac{a}{b} \right)^2
      2 = \frac{a^2}{b^2}
      2b^2 = a^2

  3. Implications:

    • The equation 2b^2 = a^2 implies that a^2 is even (since it’s 2 times an integer b^2). Therefore, a must also be even (because the square of an odd number is odd).

    • Let a = 2k for some integer k. Substitute a with 2k in the equation:

      2b^2 = (2k)^2
      2b^2 = 4k^2
      b^2 = 2k^2

    • This equation implies that b^2 is also even, which means b must be even.

  4. Contradiction:

    • If both a and b are even, they have a common factor of 2. This contradicts the original statement that \frac{a}{b} is in its simplest form (coprime).

  5. Conclusion:

    • The assumption that \sqrt{2} is rational leads to a contradiction. Therefore, the assumption is false, and \sqrt{2} must be irrational.

Final Answer:

The technique of proving a statement true by reducing it to its opposite and demonstrating the absurdity of the opposite result is known as proof by contradiction. This method is essential for establishing the validity of various mathematical statements and logical propositions, as demonstrated in the example of proving the irrationality of \sqrt{2}.