I need the working solutions and acccurate answers to this questions. Thank you.

Question

I need the working solutions and acccurate answers to this questions. Thank you.

Problem 1. (15 points) (i) For2(P(z) Q(x)), distributing the quantifier over the operator results in dieP(x) A3nQ(x). Are the two expressions equivalent? If they are, give a proof. If they are not, provide a counterexample and explain any direction of the implication that does not hold. Any counterexample needs to have a universe of size at least 100. (ii) Express the negation of 3dyR(x,y) VaVyS(x,y) so that all negation symbols immediately precede predicates. Show your work.

Explanation / Answer

i
Suppose

P(x) states that x is Even
and
Q(x)states that x is Odd

And suppose our set is Natural numbers

So if we say x (P(x) ^ Q(x))
it states there exist atleast one natural number which is Even and Odd at the same time (Which is obhiously false since no number can be odd and even at the same time)

But when we say xP(x) ^ xQ(x)
it states there exist atleast one natural number which is Even and there exist atleast one natural number (May be different from previous x) which is Odd.
And outcome of this statement is True

ii
¬(xyR(x,y) ^ xyS(x,y))
= ¬(xyR(x,y)) V ¬(xyS(x,y))
= xy¬R(x,y) V ¬(xyS(x,y))
= xy¬R(x,y) V xy¬S(x,y)

I you facing any trouble understanding the solution to the problem, please feel free to comment below. I shall get back to to you at the earliest

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