Part I: Relations 1. Represent each of the following relations using a matrix and a directed graph R is a relation on the set {1,2,3,4) such that R = {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4). Ris a relation on the set [1,2,3, 4) such that (x,y) E R if and only ifx +y>2 a. b. 2. Find the transitive closure of each of the following relations on the set {1,2,3,4) b. R {(2,1), (2,3), (3,1), (3,4), (4,1), (43). 3. Ris a relation on Z+such that (x.y) E R if and only if there is a positive integer n such thatx-y. Determine whether R is a partial order, a strict order, or an equivalence relation. Justify your answers Draw a Hasse diagram for each of the following partial orders. Find the maximal and minimal elements and determine whether the partial order is a total order. 4. a. R is a partial order on the set [0,1,2,3,4,5) such that (x,y) E R if and only if x 2 y b. R is a partial order on the set 12,3,5,10,11,15,25] such that (x, y) E R if and only if y is divisible byx. Hint: An integer y is divisible by an integer x with x y=xk. 0 if and only if there exists an integer k such that