n each case below, either give an example of the desired object, or explainwhy no such object exists. Fully explain your answers.(a) A non-empty set of irrational numbers with rational supremum. [5 marks](b) A pair of non-empty sets X ⊆ Y , where Y has an infimum and X doesnot.

(a) A non-empty set of irrational numbers with a rational supremum does exist. For example, consider the set S = {x in R: x^2 < 2}. This set is non-empty and consists of irrational numbers. The supremum of this set is sqrt(2), which is an irrational number. However, if we consider the set in the rational number system, the supremum is a rational number, which is 1.414 (approximation of sqrt(2)).

(b) A pair of non-empty sets X ⊆ Y, where Y has an infimum and X does not exist. For example, let Y = [0, 1] be a closed interval in R, and X = (0, 1) be an open interval in R. Here, Y has an infimum, which is 0, but X does not have an infimum in R because for any x in X, there exists another number in X that is less than x.