ISRO CSE 2020 | Question: 73

Question

Given that

$B(a)$ means “$a$ is a bear”

$F(a)$ means “$a$ is a fish” and

$E(a,b)$ means “$a $ eats $b$”

Then what is the best meaning of

$\forall x [F(x) \to \forall y(E(y,x)\rightarrow b(y))]$

• A. Every fish is eaten by some bear
• B. Bears eat only fish
• C. Every bear eats fish
• D. Only bears eat fish

Comments

Similar question: TIFR2017-B-11

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the formula to interpret is:

 

∀x[F(x) → ∀y(E(y, x) → B(y))]

 

First, I need to break down the formula step by step. The universal quantifier ∀x means "for all x". The formula inside is F(x) → ∀y(E(y, x) → B(y)).

 

So for every x, if x is a fish (F(x)), then something about all y. Let's look at the inner part: ∀y(E(y, x) → B(y)). This means for any y, if y eats x (E(y, x)), then y is a bear (B(y)).

 

Putting it all together: For every fish x, if any y eats x, then y is a bear. So, this translates to "Any eater of a fish is a bear" or "Only bears eat fish."

 

Now looking at the options:

A. Every fish is eaten by some bear → This would be like ∀x(F(x) → ∃y(B(y) ∧ E(y,x))), which is different because here it's saying each fish has at least one bear that eats it, but the original formula is about any eater of a fish being a bear, not necessarily that every fish is eaten.

 

B. Bears eat only fish → This would mean if a bear eats something, it's a fish: ∀y(B(y) → ∀x(E(y,x) → F(x))). But the original formula is about if something is a fish, then its eaters are bears, which is the converse.

 

C. Every bear eats fish → This would be ∀y(B(y) → ∃x(F(x) ∧ E(y,x))), but again, the original is not about bears eating fish, but about fish being eaten only by bears.

 

D. Only bears eat fish → This matches because "only bears" means if anything eats a fish, it must be a bear, which is exactly what the formula says: ∀x[F(x) → ∀y(E(y,x) → B(y))].

 

The user's thought process mentioned "For all fish there exist an animal if it eats fish then it's a bear. Only bears eat fish." This seems correct. The formula is saying that for any fish, any creature that eats that fish must be a bear. So the correct answer is D.

The logical statement ∀x[F(x)→∀y(E(y,x)→B(y))] translates to:
"For every fish x, if any entity y eats x, then y must be a bear."

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@Kshitij Sharma Thank you for this easiest explanation

Answer 1

Let us translate the given statement :

For every x,if x is a fish, then for every y, if y eats x then y is bear..

This is enforcing the condition that every animal that eats a fish is a bear.. So only option d matches..

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other options:

option a:Every fish is eaten by some bear

$\forall x(F(x)\Rightarrow\exists y(B(y)\wedge E(y,x)))$

ie. for all x, if x is a fish, then there is a y such that y is a bear and y eats x.

option b:Bears eat only fish

$\forall x(B(x)\Rightarrow\forall y (E(x,y)->F(y))$

i.e for every x, if x is a bear,then for all y ,if x eats y, then y is a fish.

option c:Every bear eats fish

$\forall x(B(x)\Rightarrow\exists y (F(y)\wedge E(x,y))$

for all x, if x is a bear, then there is a y such that, y is a fish and x eats y.

Comments

@commenter commenter

i got an example..

let the domain be {f1,f2,f3,b1,b2,e1,e2} . f1,f2,f3 are fishes, b1,b2 are bears, e1,e2 are eagles.

Here both f1,f2 are eaten by b1.

In this example, every fish is eaten by some bear is false but only bears eat fish is true.

So both the statements are not equivalent.

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@srestha can you verify these statements:

If every fish is eaten by some bear is true, then only bears eat the fish is also true.

Both options are false if at least 1 fish is eaten by some eagle.

This gives us impression​​​ that both the options are same but they are not

Correct me if iam wrong..

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Statement 1 is true then as they said are not asking about characteristic of bear they are asking what that predicate logic means. In that E(x,y) means "x eats y" so all the fish are being eaten by some bear.

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"Every fish is eaten by some bear"

means there are some bear, who eats something other than fish.

But "Only bears eat fish" mean may be "every bear eats fish."

That is the difference

Answer 2

Please refer here:

 

 

https://gateoverflow.in/99058/first-order-logic

Answer 3

From the translation of the given statement

For every x,if x is a fish, then for every y, if y eats x then y is bear.

It appears that

Every Bear eats Fish.

Hence the option C is correct.