(a) Prove that the following statement is true for all integers n: If n is even then n³ + n + 3 is odd. (b) Use proof by contradiction to show that there is no positive real number x such that Ꮖ - 2 3x Ꮖ x+2

For each of the statements below (which all happen to be false), look atthe “proofs” that have been provided and explain in one or two sentences what error has been made by the author.

Prove the following statements: (a) If n is a positive integer then n 2 + 1 is an odd positive integer. (b) If n 2 + 1 is an even positive integer, and n is a positive integer then n is an odd positive integer.

(a) Give an example of a natural number n such that n is composite, even, and √n is not a rational number. Show that each of these properties hold for your particular value of n by appealing to definitions. Do not simply provide a value of n with no justification. (b) Does your answer in (c) show that the original statement is true for all n ∈ N? Explain why or why not.