Problem 8 (16 marks) Let JB 2 {0,1} and consider the function f : IN a [B given by no :{ 3, (a) Show that for all (I, b E N: (it fit! + I7) : maxiftlfthll (ii) flab) = miniflflbll if it > 0, otherwise. From Problem 7, we know that Rf g N X li\\, the relation given by: (mm) E Rf if and is an equivalence relation. Let [E I; Powflh in] E [E denote the equivalence class of ii. We would like to define binary operations, onlyifm) =f(it) be the set of equivalence classes of Rf, and for n E N, let and - , on E as follows: x:_[y17;xlyl X_ 4 [y l: 1131]. The difficulty is that the operands [x] and y can have multiple representations (eg. if z E [x] then [I] = [2]), and so it is not Clear that such a definition makes sense: if we take a different representation of the operands, do we still end up with the sarn e result? For example, suppose [I] = [2]. Then we would want [1]E [1] : [2] E [2], but with the proposed definition above, we would have [1] [1] 7 [2] and [2] E [2] i [4 , and it is by no means clear that 2] 7 [4]- Our next step is to show that such a definition makes sense. (b) Define relations LF E E2 X F. as follows: ((X,Y),Z) El if and only if there is 1,}; E N such that X = [x], Y = [y] and Z = [I +y] ([X,Y),Z) E E if and onlyr if there is Jay E N such that X = [x], Y = [y] and Z 2 [1y] (i) Show that i is a function. (ii) Show that f: is a function. Part (b) shows that the informal definition of E and 3 given earlier is wail-dened, so from now we will View E and - as binary operations on E, that is , - Show that for all A,B,C C E: (C}AE[1]:A (d) AIR: BEA {c} AEfBEC)[AEB)E(A:CJ Remark Objects thal have a concept of \"addition" ( :lExlEHE ) and \"multiplication\" ( ' - addition and multiplication are associative, both operations have identities (see 8(0), ' addition is commutative (see Std\