Please provide workings/steps for the solutions.

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Let A and B be subsets of strings such that AB and A* are unambiguous expressions. We have learned these two rules when the weight of a string is its length: 0 AB(x) = 4(x) g(x) and Ox: (x) = (1 - 04(x))-7. For each of the following, we define a new weight function w* for a string. Determine whether or not these two rules can still be applied when we use w* as the weight function. If so, then explain why we can do so, and determine the generating series of S below with respect to w*. If not, give a general explanation of why we cannot apply one of the rules, and give an example where the rule fails. S = {1}*({000}{1, 11})*. (a) The weight of a string is the length of the string plus 2. (b) The weight of a string is the number of 1's in the string. (c) The weight of a string is the number of O's plus three times the number of 1's. Let A and B be subsets of strings such that AB and A* are unambiguous expressions. We have learned these two rules when the weight of a string is its length: 0 AB(x) = 4(x) g(x) and Ox: (x) = (1 - 04(x))-7. For each of the following, we define a new weight function w* for a string. Determine whether or not these two rules can still be applied when we use w* as the weight function. If so, then explain why we can do so, and determine the generating series of S below with respect to w*. If not, give a general explanation of why we cannot apply one of the rules, and give an example where the rule fails. S = {1}*({000}{1, 11})*. (a) The weight of a string is the length of the string plus 2. (b) The weight of a string is the number of 1's in the string. (c) The weight of a string is the number of O's plus three times the number of 1's