Consider the language = {^) where A is a function symbol of arity 2 and the...

    

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Consider the language = {^) where A is a function symbol of arity 2 and the following C-structure M. Let M be the set of strings from the alphabet {0,1} (of finite length). Let ab be the longest strings which is an initial segment of both a and b. In questions below, we will write A in place of AM to ease the notation a little. (1) For each of the following, give examples of strings a and b such that ab is (a) (b) 10 and (c) 01 (2) Give a brief explanation of why a Ab-b^a for any a.be M. (3) Find a first order C-formula with two free variables u and v such that for all a, b M, a is an initial segment of b if and only if Mo[u=a,v=b]. a, b] is a shorthand for any assignment of variables a such a and s(v) = b. Here olu that s(u) (4) Find a first order C-formula with one free variable w such that ME ofwel where e is the empty string, but for any ac, Molu=a]. Please show your work Consider the language = {^) where A is a function symbol of arity 2 and the following C-structure M. Let M be the set of strings from the alphabet {0,1} (of finite length). Let ab be the longest strings which is an initial segment of both a and b. In questions below, we will write A in place of AM to ease the notation a little. (1) For each of the following, give examples of strings a and b such that ab is (a) (b) 10 and (c) 01 (2) Give a brief explanation of why a Ab-b^a for any a.be M. (3) Find a first order C-formula with two free variables u and v such that for all a, b M, a is an initial segment of b if and only if Mo[u=a,v=b]. a, b] is a shorthand for any assignment of variables a such a and s(v) = b. Here olu that s(u) (4) Find a first order C-formula with one free variable w such that ME ofwel where e is the empty string, but for any ac, Molu=a]. Please show your work