Let P(r) and Q(r) be sentences that depend on an element z e D for some...

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Let P(r) and Q(r) be sentences that depend on an element z e D for some set D. For example, we can have D = Z, P(x) =" is even", Q(r) = is odd." Do the following sentences have the same truth value (Vx D) (P(x)⇒ Q(z)) and (vx € D. P(x)) (Vz € D,Q(x)) for every choice of D, P(x), and Q(z)? If yes, give a proof. If no, give a choice of D, P(x), and Q(z) for which the two sentences above have different truth values. 9. Let P be the sentence (VER) (²- 6x +820) (a) Use one of the generalized De Morgan's laws to show that -P is logically equivalent to (3r € R) (2²-62 +8<0) (b) Is P true or false? Hint: complete the square. 10. Write the negation of the sentence (Vr € R) (3y € R) (ry = 1). Which is true, the original sentence or its negation? Let P(r) and Q(r) be sentences that depend on an element z e D for some set D. For example, we can have D = Z, P(x) =" is even", Q(r) = is odd." Do the following sentences have the same truth value (Vx D) (P(x)⇒ Q(z)) and (vx € D. P(x)) (Vz € D,Q(x)) for every choice of D, P(x), and Q(z)? If yes, give a proof. If no, give a choice of D, P(x), and Q(z) for which the two sentences above have different truth values. 9. Let P be the sentence (VER) (²- 6x +820) (a) Use one of the generalized De Morgan's laws to show that -P is logically equivalent to (3r € R) (2²-62 +8<0) (b) Is P true or false? Hint: complete the square. 10. Write the negation of the sentence (Vr € R) (3y € R) (ry = 1). Which is true, the original sentence or its negation?

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1 The two statements do not necessarily have the same truth value Proof L... View the full answer