These are the questions on mathematical analysis

Problem la(15 marks) Prove that there exist infinitely many real number v sack that the equation 10 -trip= xy does not have any solution with x yE Q Problem 16 ( 15 marks ) Let S= Ersz: reQ}. Determine if Sis a countable set or not. Prove that there exist infinitely many real numbers c such that the equation 2 siny-2 cost= does not have any solution with x, y Es. Problem 2 ( 20marks) Let (= fatboz: a, beQ] and S= fx: KERR and x5 6x+ 7EC], Determine(with proof) whether S is countable or uncountable. Problem 3 ( 2.5 marks ) Let S be the set of order pairs (p, C), where P = (x, y ) E Q x Q and C is the circle with center p and radius / xyl+ 1. Determine if Sisq countable set or not. Problem 4 ( 2.5 marks) Let (o, = ) n REA, C [o, 1). For n = 1, 2, 3, ".., let Anti - EvX: XEAn]. Determine the supremeon and infimum of . Ax with proof