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Using the definitions of the quantifiers, it is straightforward to argue that, if Q does not contain the variable x and the universe is non-empty, then the following equivalences hold: (1) ?xP(x) ? Q ? ?x(P(x) ? Q) (2) ?xP(x) ? Q ? ?x(P(x) ? Q) Explain how these equivalences can be used to demonstrate the following: (a) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y)) (b) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y)) (c) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y)) Clearly, similar equivalences can be shown to hold when ? is replaced with ?

Name: using the definitions of the quantifiers it is straightforward to argue that if q does not contain the variable x and the universe is non empty then the following equivalences hold 1 xpx q xpx q 2 xpx
Uploaded: 2023-06-14T13:11:14-08:00
Duration: 2 min 29 s
Channel: Numerade
Description: using the definitions of the quantifiers it is straightforward to argue that if q does not contain the variable x and the universe is non empty then the following equivalences hold 1 xpx q xpx q 2 xpx

          Using the definitions of the quantifiers, it is straightforward to argue that, if Q does not contain
the variable x and the universe is non-empty, then the following equivalences hold:
(1) ?xP(x) ? Q ? ?x(P(x) ? Q) (2) ?xP(x) ? Q ? ?x(P(x) ? Q)
Explain how these equivalences can be used to demonstrate the following:
(a) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y))
(b) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y))
(c) ?xP(x) ? ?xQ(x) ? ?x?y(P(x) ? Q(y))
Clearly, similar equivalences can be shown to hold when ? is replaced with ?

Added by Mohammad A.

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