Problem 1. A bijective function from a set X to itself is sometimes called a permutation. Let X = {1,2.3}. No proof is required for this problem. (i) How many functions are there with domain X and codomain X7 How many of these functions are bijective? (1i) Let 5 denote the set of bijective functions from X to X. Describe each element of . Be sure to give each of these functions a unique name for future reference. (iii) Identify the inverses of each of these bijective functions. {iv) Compute the composition of each pair of these bijective functions in a \"multiplication\" table.! That is. draw a square |S| by |S] grid in which each row and each column is labelled with an element of 5. In each cell of the grid, compute the function composition f o g where f is the label of the row and g i= the label of the column. Here is an example of such a table: Let = {0, 1}. There are two bijective functions from to : the identity function id given by id(x) = x and the bijection f given by f(z) = 1 x. The two by two multiplication table is set up like this: id I ideid idof JlJeid fof Since idoid = id, idef = f, foid = f, and fo f = id, we can complete this table by incorporating these computations: Problem 2. A particle is travelling through the plane. The position of the particle is given by the function p : E R? defined by p(f) = (t* 1.1 t) where is time measured in seconds. (i) Let T = [2,2] be the closed interval. What is the image p[T]? Draw a picture of p[T] in the plane. (ii) According to international plane law, it is illegal for a particle to enter the region X = (oc, 0] x R with fines levied depending on the amount of time the particle spends in X. Compute the preimage p~'[X]. How many seconds did the particle spend in this region? (i) It is especially illegal for the particle to enter the region = [%._] = R. Compute the preimage p~![]. How many seconds did the particle spend in this region? {(iv) Is the function p : B R? injective? Is it surjective? For each, provide a proof or disproof