Let A = {a, b,c, d} and R = {(a, a),(a, b),(b, a),(c, c),(c, d),(d, c)}be a relation on A. Represent the given relation as a digraph. Find the matrix representation of R. Is R reflexive, irreflexive, symmetric, antisymmetric, transitive? Explain. Calculate R^2. Calculate R  R^(-1).

Need to finish my juan wants to build a greenhouse in his back yard. he needs to maximize the ratio of floor area to structure volume in order to provide the most room for plants while minimizing the area to be heated. find the description of the structure that best meets juan’s criteria. a.) a hemisphere with a diameter of 12 feet b.) a cube with a side length of 12 feet c.) a cone with a diameter of 12 feet and a height of 10 feet d.) a square pyramid with a base side length of 12 feet and a height of 9 feet