ISRO2020-73

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

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

@chirudeepnamini

what is difference between , only bear eat fish and every bear eat fish?? I havenot got it clearly.

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@srestha

Every bear eat fish: 

true when every bear eats some fish.(it may eat other animals also,but every bear must eat at least 1 fish)

false when there is a bear that doesn't eat any fish.

Only bear eat fish:

true when no animal other than bear eats fish

false if there exists a fish eaten by a animal other than bear

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yes, correct..

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srestha   chirudeepnamini

How to interpret this kind of statement.

For all x, If x is a Fish then For all y, If y eats x then y is a bear.

it sounds like every Bear Eats Fish.

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@MRINMOY_HALDER

"for all y" must have given you the impression that every bear eats fish.

only bears eat fish$:$ For all x, If x is a Fish then For all y, If y eats x then y is a bear.

look at the highlighted part 

Here it enforces the condition that every animal y that eats fish x must be a bear.

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every bear eats fish  can be viewed as " for every bear(say x) there is some fish(say y) and it(y) will be eaten by bear x"

Let me know if you have any doubt...

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Thanks.

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option a and option d appears same to me. Can you explain the difference between the 2 options? @chirudeepnamini

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@commenter commenter

even i am confused now :p..

If i get any idea, i will edit this comment and tag you..

you can look into this similar question https://gateoverflow.in/95818/tifr2017-b-11

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In option A) word given is "some bear", but it is not represented in logical statement. right?

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anyone please verify ,

option a says, There exist some bear who eats every fish.

Is it like this - $\exists x(B(x)\wedge \forall y (F(y)\rightarrow E(x,y)))$

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@MRINMOY_HALDER

your translation is true but it isn't equal to option a.

i mean:

Let the domain be{f1,f2,f3,b1,b2,b3}

where f1,f2,f3 are fishes and b1,b2,b3 are bears.

Every fish is eaten by some bear

says that f1 can be eaten by b1 or b2 or b3 and f2 can be eaten by b1 or b2 or b3 and f3 can be eaten by b1 or b2 or b3.

consider the below image.

Edge between $f_i$ and $b_i$ means $b_i$ eats  $f_i$.

 

This is one of such possibility .There are 26 other such possibilities for this domain

There exist some bear who eats every fish

says that a single bear eats all the fishes  f1,f2,f3.

there are 2 other possibilities 

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@srestha

i have represented some bear in the logical statement

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

by $\exists y B(y)$

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chirudeepnamini got it.

This is the most trickiest part in DMS

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So I feel option A is also correct. Isn't it?

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@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.