A Transition to Advanced Mathematics by Smith, et.al (7 th ed.) Hint for § 1.3 Exercise 7 Theorem 1.3.1 : Let A ( x ) be an open sentence with variable x . (a) ∼ ( ∀ x ) A ( x ) is equivalent to ( ∃ x ) ∼ A ( x ) ∼ [( ∀ x ) A ( x )] is equivalent to ( ∃ x )[ ∼ A ( x )] (b) ∼ ( ∃ x ) A ( x ) is equivalent to ( ∀ x ) ∼ A ( x ) ∼ [( ∃ x ) A ( x )] is equivalent to ( ∀ x )[ ∼ A ( x )] Recall the Deﬁnition : Two logical forms of quantiﬁed sentences are equivalent iﬀ the truth of one implies the truth of the other, and conversely, for every possible meaning of the open sentences in every universe. Proof of Theorem 1.3.1a as given in class. Let U be any universe. Let A ( x ) be an open sentence with variable x ∈ U . It is enough to show that ∼ [( ∀ x ) A ( x )] is true in U if and only if ( ∃ x )[ ∼ A ( x )] is true in U . Note that TFAE. < TFAE = the following are equivalent. > < Steps marked by * are skipped in the book. > • ∼ [( ∀ x ) A ( x
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