Proof：練習問題

A student attempts to prove that for all integers \(n\), the expression \(n^2 - n + 41\) is a prime number. Which value of \(n\) provides a counter-example to this statement?

Identify the error in the following deductive proof that \(2 = 1\):
1. Let \(a = b\)
2. \(a^2 = ab\)
3. \(a^2 - b^2 = ab - b^2\)
4. \((a - b)(a + b) = b(a - b)\)
5. \(a + b = b\)
6. \(2b = b\)
7. \(2 = 1\)

A student is using proof by exhaustion to show that \(n^2 + 2\) is not divisible by 4 for any integer \(n\) such that \(1 \le n \le 3\). Which set of calculations completes this proof?

To disprove the statement "\(2^n - 1\) is a prime number for all odd integers \(n > 1\)", which value of \(n\) should be used as a counter-example?

A student is asked to prove that for all real values of \(x\), \(x^2 - 4x + 7 > 0\).
Which of the following algebraic steps is the most appropriate starting point for a proof by deduction?

Show by counter-example that the statement \( |x + y| = |x| + |y| \) is not true for all real numbers \( x \) and \( y \).

Use the method of deduction to prove that for all real values of \( x \), the quadratic expression \( x^2 - 4x + 7 \) is always positive.

A student claims that \( n^2 + n + 11 \) is a prime number for all positive integers \( n \). Provide a counter-example to disprove this statement.

Prove by deduction that the product of two odd numbers is always odd.

(a) Use the method of exhaustion to show that \( n^2 - n + 11 \) is a prime number for all integers \( n \) where \( 1 \le n \le 3 \).
(b) Show that the statement "\( n^2 - n + 11 \) is prime for all positive integers \( n \)" is false by finding a counter-example.