Consider an alphabet consisting of only three letters m, i and u, together with the following rules for creating new strings from old ones. (you can apply the rules in any order.) rule 1. if you have a string whose last letter is t, then you can add a u at the end of the string. for example, mi can be transformed into miu, and ii can be transformed into in using rule 1. rule 2. if the initial is m, then the set of all letters after the m can be doubled. in symbols, mx becomes mxx where x is a variable that can consist of any string. here are some examples of transformations using rule 2: • min is transformed into mini • mum is transformed into mumm • mu is transformed into muu rule 3. if it occurs in a string. then you can replace it with u. here are some examples of transformations using rule 3: • umitimu is transformed into umumu • mit is transformed into mui (or alternatively min). iimii can't be transformed using rule 3 because the three is are not consecutive. • mill is transformed into mu rule 4. if uw occurs in a string, then you can drop it. for example, uuu can be transformed into it, and muuuii can be transformed into muili using rule 4. also, note that the rules can be applied only in one direction. for example, we can use rule 3 to transform mill into mu, but you can't use rule 3 to convert mu into miii. (a) prove: if mi, then mutiu (b) prove: if mi, then mui. 2. a rational number is a number that can be written as a fraction of integers. more precisely, z is rational if x = -, where p and q are integers and q +0. prove: the sum of two rational numbers is a rational number.