Let f : X → Y and g: Y → Z be two maps. (a) Show that if g o f is surjective, then g must be surjective. (b) Show that if g o f is injective, then f must be injective.

Determine a mapping of the form (x, y) → (, ) for the inverse of a relation. a. (x, f(x)) b. (x, y¹) c. (y, f(x)) d. (y, x)

Let M(R) denote the set of all mappings from R to R. For mappings f, g: R → R, define pointwise addition f + g of by (f+g)(x) = f(x) + g(x), Vx € R, and pointwise multiplication by (f g)(x) = f(x) · g(x), Vx R. (a) Verify that the pair (M(R), +) form a group. (b) Verify that the pair (M(R), .) does NOT form a group. Suggest a modification to the underlying set M(R) (you may call this new set M(R)#), so that the pair (M(R)#, .) would form a group. E (c) Define a set H(R) := {f = M(R) : f(x) = Z, for each x € R}. Prove that the pair (H(R), +) is a group.

(a) Construct a linear map L(z)=az+b such that L(1)=i and L(i)=1 (b) Is the map you constructed in part (a) unique? Please Explain.