Question: 1. First Order Formulas and Interpretations We fix a signature S = (c1, f1(), f2(, ), f3(, ), R1(, ), R2(, )) and consider the

1. First Order Formulas and Interpretations We fix a signature S = (c1, f1(), f2(, ), f3(, ), R1(, ), R2(, )) and consider the interpretation I = (Z, 0, x 7 x + 1,(x, y) 7 x y,(x, y) 7 x + y, , =) (as on slide 5 of Lecture 21). (a) Translate the following formulas to English/Math: 1 = (x(y(R1(y, f3(x, x))))) 2 = (y(x(R1(y, f3(x, x))))) 3 = (y(x(R1(y, f2(x, x))))) 4 = (x(R2(f1(x), f3(x, y)))) 5 = ((R1(f1(c1), x)) (R1(f3(x, x), x))) (b) For each of the above formulas, state whether or not they are satisfied in interpretation I. For formulas that are not sentences (that is, have free variables) provide a valuation v such that I, v together satisfy the formula or argue that no such valuation exists. (c) For each of the following sets of numbers, provide a formula that defines this set in interpretation I: The set that contains only the number 0 The set of prime numbers 1. First Order Formulas and Interpretations We fix a signature S =

1. First Order Formulas and Interpretations We fix a signature and consider the interpretationn (as on slide 5 of Lecture 21) (a) Translate the following formulas to English/Math b) For each of the above formulas, state whether or not they are satisfied in interpre- tation I. For formulas that are not sentences (that is, have free variables) provide a valuation v such that I,v together satisfy the formula or argue that no such valuation exists c) For each of the following sets of numbers, provide a formula that defines this set in interpretation The set that contains only the number 0 . The set of prime numbers 1. First Order Formulas and Interpretations We fix a signature and consider the interpretationn (as on slide 5 of Lecture 21) (a) Translate the following formulas to English/Math b) For each of the above formulas, state whether or not they are satisfied in interpre- tation I. For formulas that are not sentences (that is, have free variables) provide a valuation v such that I,v together satisfy the formula or argue that no such valuation exists c) For each of the following sets of numbers, provide a formula that defines this set in interpretation The set that contains only the number 0 . The set of prime numbers

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