prove or disgrove. = (c) Is the formula true in the case A = Q? Prove or disprov

Question

prove or disgrove. = (c) Is the formula true in the case A = Q? Prove or disprove. 3. Consider the formulas A: Vr.P(z) Vr.Q(z): B : Vr.(P(z) Q(z)). (a) Is A B a tautology? (b) Is B A a tautology? (c) Are A and B logically equivalent? THREE No! Counter Esample Demain:N A is true, but B is false! Hence, A e is NOT a tautalJY Q: prime Cb) Is A a tautoelegy Assume B is true Tis means fot all tn the domain (an liment) p(c) Q(L) (s true. Hese, for all e where Pcc) is trse and QCc) ts tr To prove A, reason oy cases Case Pce halds PC is tre Hence, in either case we have shown that BA ls a tautalo Ce A andl B are NoT dogicalty equivalant becaus

Explanation / Answer

The solution is correct .
the following example is enough to prove your problem.it take sthe 2 cases into 1 case;

In the example

x(P(x)Q(x))xP(x)xQ(x)(1)(1)x(P(x)Q(x))xP(x)xQ(x)

we have the following different first order formulas:

These will be replaced by A,BA,B, and CC, respectively. We get

ABCABC

which is not a tautology; its truth value depends on the truth values of the variables.

(1)(1) would be a tautology of first order logic if we could create it by the reverse substitution from a tautology of propositional logic.

For another example take the propositional tautology

ABABABAB

and let's do the following substitutions:

Now, we have

x(yQ(x,y))yP(y)x(yQ(x,y))yP(y)x(yQ(x,y))yP(y)x(yQ(x,y))yP(y)

which is a tautology of the first order logic because it could be obtained by a substitution of first order expressions into a propositional tautology.

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