Math Problem Statement

The Foundations: Logic and Proofs create a example of logic proof

Solution

Example of a Logic Proof: Proof by Contradiction

Problem Statement:

Prove that the square root of 2 is irrational, i.e., 2\sqrt{2} cannot be expressed as a ratio of two integers.

Proof:

We'll use a proof by contradiction. The idea is to assume the opposite of what we want to prove and then show that this assumption leads to a contradiction.

  1. Assumption (Opposite of What We Want to Prove):
    Assume 2\sqrt{2} is rational. This means that 2\sqrt{2} can be expressed as a fraction: 2=ab\sqrt{2} = \frac{a}{b} where aa and bb are integers with no common factors (other than 1), and b0b \neq 0.

  2. Square Both Sides:
    Squaring both sides of the equation gives: 2=a2b22 = \frac{a^2}{b^2} Multiply both sides by b2b^2 to eliminate the fraction: 2b2=a22b^2 = a^2

  3. Implication for a2a^2:
    This equation shows that a2a^2 is even (since it equals 2 times something). If a2a^2 is even, then aa must also be even (because the square of an odd number is odd).

  4. Express aa as an Even Number:
    Since aa is even, we can write a=2ka = 2k for some integer kk.

  5. Substitute a=2ka = 2k into the Equation:
    Substitute into the equation 2b2=a22b^2 = a^2: 2b2=(2k)2=4k22b^2 = (2k)^2 = 4k^2 Simplifying, we get: b2=2k2b^2 = 2k^2 This equation shows that b2b^2 is even, which means bb is also even.

  6. Contradiction: Now we have shown that both aa and bb are even, which means that both aa and bb have a common factor of 2. However, this contradicts our original assumption that aa and bb have no common factors other than 1.

  7. Conclusion: Since our assumption that 2\sqrt{2} is rational leads to a contradiction, we must conclude that 2\sqrt{2} is irrational.

This concludes the proof.


Would you like more details on any step, or do you have any questions?

Here are 8 related questions:

  1. What is the difference between proof by contradiction and proof by contrapositive?
  2. How can you prove that other square roots, like 3\sqrt{3}, are irrational?
  3. Can you provide an example of a direct proof?
  4. What is the importance of logic in mathematical proofs?
  5. How do you prove a statement using proof by induction?
  6. What are the key differences between different types of proofs?
  7. How can you prove that an infinite number of primes exist?
  8. Can you explain what a constructive proof is?

Tip: When working with proofs, always start by clearly stating your assumptions and what you need to prove. This helps guide your logic and ensures clarity.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Logic
Proofs
Number Theory

Formulas

-

Theorems

Proof by Contradiction

Suitable Grade Level

Advanced High School and above