CHAPTER Review Exercises 5 Discuss the truth of the following statements. Prove the ones that are true; find a counterexample for each one of the false statements, and follow directions. The exercises with the symbol (*) require knowledge of calculus and/or linear algebra. 1. If P(x 1 , y 1 ) and Q(x 2 , y 2 ) are two distinct points in the plane, then the distance between the two of them, defined as q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dðP, QÞ = ðx 2 - x 1 Þ 2 + ðy 2 - y 1 Þ 2 , is a positive number. 2. Let a be a real number. Then the opposite of a is unique. 3. If n is any positive integer number, then ln n < n. Prove this statement in all of the following ways: a. By induction.