Similar question: TIFR2017-B-11

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7 votes

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

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

- Every fish is eaten by some bear
- Bears eat only fish
- Every bear eats fish
- Only bears eat fish

10 votes

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..

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.

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.

@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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