(0)f(x)=1/(1-x)-x².

(a)Let e, be the number of edges in the Hasse diagram for the partial order. Please find a recurrence relation for and then solve the recurrence relation to get

(b) Please find the number of vertices of the diagram in (a).

(a)Find all generators of the cyolic group (Z8 +).

(b)Find all generators of the cyelic group (Z5-{0}, *).

(c) If G is a cyclic group of order n, how many distinct generators does it have?

9. Please find an encryption function E: {1, 2, 3, ... 26} -＞ Z such that E(m1) + E(m2) = E(m1+m2) for every m1, m2 in {1, 2,3, ... ., 26} and please also find the decryption function corresponding to E.

(1)

3. Two-way insertion sort : Suppose the output sorted sequence is increasing. The two-way insertion sort is a modification of the simple insertion sort (straightforward insertion sort), described as follows. After read the input elements, a separate output array a[0], a[1], a[2], a[n-1] is used to store the sorted sequence. This output array acts as a circular structure, that is, the right position of a[i] is a[i+1] if 0≤ i ≤ n-2, and the right position of a[n-1] is a[0]. The first input element is put into a[0] initially. Once a contiguous group of elements are in the array, room for a new input element is made by shifting all smaller  elements one step to the left or all larger element one step to the right. The choice of left-shift or right-shift to perform depends on which would cause the smallest amount of shifts. Use the following 6 input elements to illustrate how this algorithm works. You have to show the content of the output array after each input element is inserted into the array. 27,35,43, 31,29,33.

(1) The label of each leaf node is a single symbol. The label of each internal node is the concatenation of the labels of its left child and right child.

(2) When two nodes are merged, always set the node of label with less lexical order as the left child, and the other as the right child.

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