1. Using proof by contrapositive,show that if a andb are integers and √3 is rational, then b/a ≠ √3 [15 pts]
2. Using proof by contradiction,show that there is no greatest integer. [15 pts]
3. Using proof by contradiction,show that if n² is divisible by 4,then n is divisible by 2.[15 pts]
4. Suppose n ∈ Z.Prove the following statement: 5n + 3 is odd if and only if n is even.[15 pts]
5. For each of the following statements,determine whether it is true or false. If true, provide a proof. If false disprove it.
(a)Suppose x ∈ R. Then |x| = 3 if and only if x = 3 or x = -3. [5 pts]
(b)Suppose x ∈ R. Then x² = 9 if and only if x = 3. [5 pts]
6. Prove or disprove the following statement:For any two different rational numbers,there's another rational number strictly between them. [15 pts]
7. Prove by contradiction that 6x³ + 8x + 1 = 0 has no integer solutions. [15 pts]