1. The problem asks why assuming the opposite of what we want to prove helps reveal hidden truths in mathematics, using the proof that $\sqrt{2}$ is irrational as an example. 2. Proof by contradiction starts by assuming the negation of the statement we want to prove. This assumption often leads to logical consequences that contradict known facts or properties, revealing that the original assumption must be false. 3. For example, to prove $\sqrt{2}$ is irrational, we assume the opposite: that $\sqrt{2}$ is rational. This means it can be expressed as a fraction $\frac{a}{b}$ in lowest terms, where $a$ and $b$ are integers with no common factors. 4. Squaring both sides gives: $$\sqrt{2} = \frac{a}{b} \implies 2 = \frac{a^2}{b^2} \implies a^2 = 2b^2$$ 5. This implies $a^2$ is even, so $a$ must be even (since the square of an odd number is odd). Let $a = 2k$ for some integer $k$. 6. Substitute back: $$a^2 = (2k)^2 = 4k^2 = 2b^2 \implies 2k^2 = b^2$$ 7. This shows $b^2$ is even, so $b$ is also even. 8. But if both $a$ and $b$ are even, they share a factor of 2, contradicting the assumption that $\frac{a}{b}$ is in lowest terms. 9. This contradiction means our initial assumption that $\sqrt{2}$ is rational is false, so $\sqrt{2}$ is irrational. 10. This example shows how assuming the opposite leads to discovering impossible properties (both numerator and denominator even in lowest terms), revealing hidden truths about numbers. 11. The second problem asks to explain the difference between syntactic and semantic errors in proofs and which is harder to detect. 12. A syntactic error is a mechanical or algebraic mistake, such as incorrect manipulation of equations or arithmetic errors. 13. A semantic error involves misapplying a theorem or misunderstanding its domain, such as using the Fundamental Theorem of Arithmetic (which applies only to integers) in a non-integer or field setting. 14. Semantic errors are harder to detect because the steps may look algebraically correct, but the logic or assumptions are invalid or inapplicable. 15. Syntactic errors are often easier to spot because they violate formal rules or produce obvious inconsistencies. 16. Therefore, semantic errors require deeper understanding of the concepts and conditions behind the theorems, making them more subtle and challenging to identify. Final answers: - Assuming the opposite in proof by contradiction helps reveal contradictions that prove the original statement. - Semantic errors are harder to detect than syntactic errors because they involve conceptual misunderstandings rather than mechanical mistakes.