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

### Example of a Logic Proof: **Proof by Contradiction**

#### Problem Statement:
Prove that the square root of 2 is irrational, i.e., \( \sqrt{2} \) 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 \( \sqrt{2} \) is rational. This means that \( \sqrt{2} \) can be expressed as a fraction:
   \[
   \sqrt{2} = \frac{a}{b}
   \]
   where \( a \) and \( b \) are integers with no common factors (other than 1), and \( b \neq 0 \).

2. **Square Both Sides:**  
   Squaring both sides of the equation gives:
   \[
   2 = \frac{a^2}{b^2}
   \]
   Multiply both sides by \( b^2 \) to eliminate the fraction:
   \[
   2b^2 = a^2
   \]

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

4. **Express \( a \) as an Even Number:**  
   Since \( a \) is even, we can write \( a = 2k \) for some integer \( k \).

5. **Substitute \( a = 2k \) into the Equation:**  
   Substitute into the equation \( 2b^2 = a^2 \):
   \[
   2b^2 = (2k)^2 = 4k^2
   \]
   Simplifying, we get:
   \[
   b^2 = 2k^2
   \]
   This equation shows that \( b^2 \) is even, which means \( b \) is also even.

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

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

This concludes the proof.

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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 \( \sqrt{3} \), 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.

Explore a detailed example of a logic proof demonstrating the irrationality of the square root of 2. This proof uses the technique of proof by contradiction, assuming the opposite and deriving a contradiction. Suitable for advanced high school students and above.

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