GATE2016-2-GA-08

Question

All hill-stations have a lake. Ooty has two lakes.

Which of the statement(s) below is/are logically valid and can be inferred from the above sentences?

1. Ooty is not a hill-station.
2. No hill-station can have more than one lake.

• A. (i) only.
• B. (ii) only.
• C. Both (i) and (ii)
• D. Neither (i) nor (ii)

Answer 1

All hill stations have a lake  $\Rightarrow  \forall x( h(x)\to (\exists y, l(y) \wedge has(x,y)))$

Ooty has two lakes $\Rightarrow \exists x ( o(x) \wedge \exists y,z ( has(x,y,z)\wedge l(y)\wedge l(z)\wedge (z!=y) )$ 

Here, $h(x) \to x \text{ is hill station}$

• $l(x) \to x \text{ is lake }$
• $has(x,y)\to x \text{ has }y$
• $has(x,y,z)\to x\text{ has } y, z$
• $o(x)\to x \text{ is Ooty}$

1. Ooty is not a hill station $\implies$ we can not derive this above arguments, Ooty has two lakes already, if Ooty had $0$ lakes only then this can become true.
2. No hill station can have more than one lake

All arguments here are saying are if we have hill station, it can have lake. It is nowhere told that how many lakes it has. So, this is false.

Answer: (D) neither (i) nor (ii)

Comments

Also, there could exist some places which have lakes but those place might not be a hill station.

$(could\ be\ True \ or\ false)?\rightarrow Has\ a\ lake(give\ as\ true ) \\ don't\ goes\ backward\leftarrow True$

only false goes backward.

so we can't say if a place has "two lakes"(basically "having a lake" condition is true) is a hill station or not because both are possible.$$0\ 0\ T \\{\color{Green} 1}\ {\color{Red} 1}\ T \\1\ 0\ F \\{\color{Yellow} 1}\ {\color{Red} 1}\ T$$

Answer 2

Every hill station has a lake means "every hill station has at least one lake". That means a hill station can have more than one lakes. But, a place without any lake cannot be a hill station. So, none can be concluded.