GATE CSE 2004 | Question: 23, ISRO2007-32

Question

Identify the correct translation into logical notation of the following assertion.

Some boys in the class are taller than all the girls

Note: $\text{taller} (x, y)$ is true if $x$ is taller than $y$.

• A. $(\exists x) (\text{boy}(x) \rightarrow (\forall y) (\text{girl}(y) \land \text{taller}(x, y)))$
• B. $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \land \text{taller}(x, y)))$
• C. $(\exists x) (\text{boy}(x) \rightarrow (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$
• D. $(\exists x) (\text{boy}(x) \land (\forall y) (\text{girl}(y) \rightarrow \text{taller}(x, y)))$

Comments

(A) (∃x) (boy(x) → (∀y) (girl(y) ∧ taller(x,y))) 
(B) (∃x) (boy(x) ∧ (∀y) (girl(y) ∧ taller(x,y))) 
(C) (∃x) (boy(x) → (∀y) (girl(y) → taller(x,y))) 
(D) (∃x) (boy(x) ∧ (∀y) (girl(y) → taller(x,y)))

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For those who are still confused between ^ and ->.

Use ^ for there exist(∃) and use -> for all(∀).

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@vishal there exist operator is distributed over OR not AND so that it gives values according to the domain.

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Can somebody please recommend me some good ( and short) resource which will be helpful for the predicate logic problems? I am really stuck in this topic.

Answer 1

Now many people get confused when to use $\wedge$ and when to use $\implies$. This question tests exactly that.
We use $\wedge$ when we want to say that the both predicates in this statement are always true, no matter what the value of $x$ is. We use $\implies$ when we want to say that although there is no need for left predicate to be true always, but whenever it becomes true, right predicate must also be true.
Now we have been given the statement $\text{ “Some boys in the class are taller than all the girls"}$. Now we know for sure that there is at least a boy in class. So we want to proceed with $ “\left(\exists x\right)(boy\left(x\right)\wedge"$ and not $ “\left(\exists x\right) (boy\left(x\right) \implies"$, because latter would have meant that we are putting no restriction on the existence of boy i.e. there may be a boy-less class, which is clearly we don't want, because in the statement itself, we are given that there are some boys in the class. So options (A) and (C) are ruled out.
Now if we see option (B), it says, every y in class is a girl i.e. every person in class is a girl, which is clearly false. So we eliminate this option also, and we get correct option (D). Let us see option (D)explicitly also whether it is true or not. So it says that if person y is a girl, then x is taller than y, which is really we wanted to say.
So option (D) is correct.

http://www.cse.iitd.ac.in/~mittal/gate/gate_math_2004.html
 

Comments

wow explaination..great work

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This is best explaination soo far in FOL..much appreciated..thankyou

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But what if there is no no girl, option D will still give true which shouldnt be.

Answer 2

I have attached two links here. They contain photos clicked from my worked out solution. Hopefully this will clear the doubts arising between option B and D.

https://i.stack.imgur.com/zc2j5.jpg

https://i.stack.imgur.com/hgbnR.jpg

Comments

Thanks...very good explanation :)

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what a explanation mini bro :) superrb

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@minipanda you explained amazingly well sir.

Answer 3

(∃x)(boy(x) ⋀ (∀y)(girl(y) → taller(x, y)))

Should be the answer.

Comments

I understod the solution but just wanna ask one thing for extra understanding. Can we move (∀y) ie for all y just before the inner paranthesis that is just after ∃x, according to null quantification rule??

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ya sushmita

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But what if there is no no girl, option D will still give true which shouldnt be.

Answer 4

we have been given the statement "Some boys in the class are taller than all the girls". Now we know for sure that there is atleast a boy in class. So we want to proceed with "(∃x) (boy(x) ∧" and not "(∃x) (boy(x) →", because latter would have meant that we are putting no restriction on the existence of boy i.e. there may be a boy-less class, which is clearly we don't want, because in the statement itself, we are given that there are some boys in the class. So options (A) and (C) are ruled out.

Now if we see option (B), it says, every y in class is a girl i.e. every person in class is a girl, which is clearly false. So we eliminate this option also, and we get correct option (D). So it says that if person y is a girl, then x is taller than y, which is really we wanted to say.
So option (D) is correct.

Comments

But the question says taller than all girls that means there is no question on existence of girls also. Doesn't it? Then it should make (B) right answer??

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But what if there is no no girl, option D will still give true which shouldnt be.