6. For the toy models in this problem, the universes will be sets of real numbers,...

 

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6. For the toy models in this problem, the universes will be sets of real numbers, and they will use the following definitions: Z(x): P(x): "x is an integer." "x is positive." G(x, y): CCC 'x is greater than y." (Specifically x > y, not x y.) (a) Show that VxVyG(y, x) is satisfiable. (In other words, create a model that satisfies this formula.) Hint: If you feel like this is impossible, maybe your universes are too small... How big of a universe can you make? (b) Is {\x(Z(x)P(x)), 3x (Z(x)^P(x))} consistent? If your answer is yes, justify your answer by giving a toy model that satisfies both formulas in the set. If your answer is no, explain why no such model exists. (c) Given the above definitions, is there a toy model that satisfies xy(G(x,y) \ G(y, x))? If your answer is yes, give a toy model that satisfies the formula. If your answer is no, explain why no such toy model exists. (d) Bonus: What does your answer to the previous question say about whether or not Exy(G(x, y) \ G(y, x)) is satisfiable? Justify your answer with examples and/or explanations as appropriate. 6. For the toy models in this problem, the universes will be sets of real numbers, and they will use the following definitions: Z(x): P(x): "x is an integer." "x is positive." G(x, y): CCC 'x is greater than y." (Specifically x > y, not x y.) (a) Show that VxVyG(y, x) is satisfiable. (In other words, create a model that satisfies this formula.) Hint: If you feel like this is impossible, maybe your universes are too small... How big of a universe can you make? (b) Is {\x(Z(x)P(x)), 3x (Z(x)^P(x))} consistent? If your answer is yes, justify your answer by giving a toy model that satisfies both formulas in the set. If your answer is no, explain why no such model exists. (c) Given the above definitions, is there a toy model that satisfies xy(G(x,y) \ G(y, x))? If your answer is yes, give a toy model that satisfies the formula. If your answer is no, explain why no such toy model exists. (d) Bonus: What does your answer to the previous question say about whether or not Exy(G(x, y) \ G(y, x)) is satisfiable? Justify your answer with examples and/or explanations as appropriate.