PREDICATE LOGIC DOUBT

Question

Only Area 51 has Extra-Terresstrials

A(x) = x is Area 51

E(x) = x has Extra-Terresstrials

Which of the following is correct?

1. (∀x)(A(x) -> E(x))

2. (∀x)(E(x) -> A(x))

3. (∀x)(A(x) <-> E(x))

Comments

I think that option 3 is the correct answer.

Only area 51 has extra terrestrials means:

If it is area 51,then it has extra terrestrials and if an area has extra terrestrials then it is area 51( since it is said that only area 51 has extra terrestrials)

So this boils down to A(X)->E(X) and E(X)->A(X),for all X. Hence option 3 is correct.

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Answer given is Option 2

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But there is no way we can make option 1 as invalid. If it is true that it is area 51,then it means that it has extra terrestrials. So both the options can't be made invalid. Hence option 3 should have been right..what is the explanation provided?

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Ok I think I got my mistake..

"only area 51 has extra terrestrials" means that if extra terrestrials are present, they can be present only in area 51..but if the area is 51,then that doesn't imply anything about the extra terrestrials.. So for all X,  E(X)->A(X) and not vice versa.. So option 2 is correct.

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But why for all is used

I think there exists is more relevent$\exists x(E(x)  \Lambda A(x))$

 

 

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@srestha ma'am, Here it is mentioned in the question that only area 51 has extra terrestrials.. So if extra terrestrials is present in any area, then it can be present only in area 51, so for all has to be used. Because if we use there exists, then the predicate will become true even if extra terrestrials is present in an area that isn't area 51 as well as if extra terrestrials  is present in the area 51.

Answer 1

This is logically equivalent to saying "If x is an extra terrestrial, it is present in Area 51."

Therefore, it can be given as $\forall x (E(x) \rightarrow A(x))$.

Comments

@Balaji Jegan

@goxul if E(x) becomes false means x has not extra terrestrial ,then LHS of implication becomes false & overall it becomes true irrespective of RHS!

which means if x has not extra terrestrial then x is area 51 is true

please clear the doubt!