1. (18 points) Take the set S to be {0, 1,2,3,4,5,6,7,8,9,10}. For each question below, answer either yes or no (2 points) and then justify your answer (4 points) by providing a counter-example if you answered no or a proof if you answered yes. (a) Is the function d(a, b) = sin(| a -b) (with both a,b in S) a metric on S? 0, if a = b, d(a, b) = 1, if a = b+1, (b) Does the function 3, otherwise (with both a.b in S) satisfy the triangle inequality axiom for a metric on S? (c) Suppose that i (a, b) and d2 (a, b) are both metrics on the set S. Will the function d(a, b) = d, (a, b) + 2d2 (a, b) also be a metric on the set S