Predicate Logic question?

Question

Find a model for every phrase where the phrase is not true:
(1) Ax Ey Pxy → Ey Ax Pxy
(2) Ax (Qx or Rx) → (Ax Qx or Ax Rx)
(3) (Ax Sx → Ax Tx) → Ax (Sx → Tx)

(A = for all, E = there exists)

Answer 1

OK, I'll give it a shot.

1) Let x be a member of the set of all states in the USA, y be a member of the set of all cities in the USA, and P(x, y) be "y is the capital of x".

The first part would be read as:

"For all states, there exists a city such that the city is the capital of the state."

Or equivalently, "Every state has a capital city."

The second part would be read as:

"There exists a city such that, for all states, the city is the capital of the state."

That's like saying "There is a city that is the capital of every state."

That's clearly false. (Washington, DC doesn't count; it's the capital of the nation, not the capital of every state.)

2) Let x be a member of the set of all people. Q(x) is "x is male" and R(x) is "x is female".

Then the first sentence reads "For each person, that person is male or that person is female." Or less awkwardsly, "Everyone is male or female." (Which is true, genetically at least.)

The second sentence reads "Every person is male or every person is female." The first part is false, the second part is false, therefore the OR of the two parts is false.

3) This one is a lot trickier.

The first sentence says "If for all x, S(x), then for all x, T(x)." The second one says "For all x, if S(x) then T(x)." There's a very very subtle difference there.

Let x be a member of the set of all seniors at a Podunk high school.

Let S(x) be "x graduates", and let T(x) be "x is a member of the first class of Podunk High where every senior graduated."

The first sentence says:

"If every senior graduates, then every senior is a member of the first class of Podunk High where every senior graduated."

The second sentence says:

"For every senior, if that senior graduates, he or she is a member of the first class of Podunk High where every senior graduated."

Do you see the difference there?

The first sentence is true because, if even one senior doesn't graduate, the hypothesis is false and the sentence is true by default. If every senior DOES graduate, the sentence is still true of course.

The second sentence is not always true, because if Joe doesn't graduate, then the sentence doesn't apply to him (the hypothesis is false), but if Mary graduates, the sentence is false for her (because there are members of the class who didn't graduate).

There's probably a better example, but that's the best one I could think of.

Answer 2

(3) ( ax sx ) - ( ax tx ) this is a radio transmitter i believe .

Answer 3

what is p, q, r ?????