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(i) Look at the following sentences written in first-order logic

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(i) Look at the following sentences written in first-order logic. Explain these sentences in plain English. Based on the definitions on validity and satisfiability, are these statements valid? If not, are they satisfiable? 

 

∀x ∃y Loves(x, y) ⇔∃x ∀y Loves(x, y) 

∀x Loves(x, movie) ⇔ ¬∃x ¬Loves(x, movie) 

∃x Loves(x, movie) ⇔ ¬∀x ¬Loves(x, movie) 

¬∀x ¬ Loves(x, movie) ⇔ ∀x Loves(x, movie) 

 

 

(ii) Formalize the following statements. Use the deduction proof in which the first three statements are facts, and the last line is the conclusion: 

 

• Every young and healthy person likes football. 

• Every active person is healthy. 

• Some ASU students are young and active. 

• Therefore, some ASU students like football. 

 

Use Y(x) for "x is young," H(x) for "x is healthy," A(x) for "x is active," B(x) for "x likes football," and N(x) for "x is an ASU student". 

 

Note: this simple question can also be proved by forward/backward reasoning. However, you are required to use refutation resolution.