4. Let I={0, 1} be the input alphabet and O = {0, 1} be the output aplphabet of a finite state machine that recognizes all strings in the language {0, 1}* {01} {0, 1}⁺. Please draw a state diagram for the finite state machine after minimization.

(a)

6. Consider the number of the partitions of a positive integer into positive summands without regard to order. For example, 4 = 1+1+1+1 = 1+1+2 = 2+2 = 1+3, so the integer 4 has five partitions. Find the generating function for the number of the partitions of a positive integer n into positive summands where each summand appears at most twice in each of the partitions. Please express your generating function in a closed form and indicate how to find the number of the partitions of n from your generating function.

7. Please compute , and y= ((w-x) mod 17), and then output (w, x, y).

(a) Please show that () is a commutative group. 

(b) What is the Principle of Mathematical Induction?

(c) Please use the Well-Ordering Principle to prove that the Principle of Mathematical Induction is true?

(b)(x7+x8+x9+...)9；

(c)

(b) Prove that () is isomorphic to ().

1. Solve the general and particular solutions of the following ordinary differential equation(ODE).y'cos2 x+3y=1,

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