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Background A word over an alphabet of symbols is a string 010203. - of symbols from . For example, if I = {a,b,c}, then the following are words over among many others: cbaca. ba. acbbca. a (which denotes the empty word). Let 2* be the set of all words over (which includes e, the empty word). Associated with each word w e * is a set of positions. For example, {1,2,3} is set of positions of the word abe with the symbol a occupying position 1, b occupying position 2, and c occupying position 3. If u, V E *, w is the word in * that results from concatenating u and v. For example, if u = aba and v=bba, then w=ababba. A language L over is a subset of S*. A language can be specified by a first-order formula in which the quantifiers range over the set of positions in a word. In order to write such formulas we need some predicates on positions. last(*) asserts that position 1 is the last position in a word. For HEL, (2) asserts that the symbol a occupies position 1. For example, the formula Vr. last(3) +(2) says the symbol a occupies the last position of a word. This formula is true, e.g., for the words aaa, 2, and bea. The language over specified by a formula is the set of words in S* for which the formula is true. For example, if a , then Vr. last (2) + a(z) specifies the language {ua u *}. 1. (12 points) Let I be a finite alphabet and I* be the set of words over 2. Define u