in Mathematical Logic edited by
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32 votes
32 votes

Given that

  • $B(x)$ means "$x$ is a bat",
  • $F(x)$ means "$x$ is a fly", and
  • $E(x, y)$ means "$x$ eats $y$",

what is the best English translation of $$ \forall x(F(x) \rightarrow \forall y (E(y, x) \rightarrow B(y)))?$$

  1. all flies eat bats
  2. every fly is eaten by some bat
  3. bats eat only flies
  4. every bat eats flies
  5. only bats eat flies
in Mathematical Logic edited by
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4 Comments

For all x if its a fly then for all y if y eats x then y is a bat ... option E ..
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Interesting fact is this question repeated in ISRO 2020 (only bears eat fish)
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1. All flies eat bats:
∀x (F(x) → ∃y (Eats(x, y) ∧ B(y)))

 2. Every fly is eaten by some bat:
∀x (F(x) → ∃y (Eats(y, x) ∧ B(y)))

 3. Bats eat only flies:
∀x (B(x) → ∀y (Eats(x, y) → F(y)))
or
∀x∀y (B(x)∧Eats(x, y)→F(y))

 4. Every bat eats flies:
∀x (B(x) → ∃y (Eats(x, y) ∧ F(y)))

 5. Only bats eat flies:
∀x∀y (F(x) ∧ Eats(y, x)  → B(y))
or
∀x (F(x)→∀y (Eats(y, x) → B(y)))

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3 Answers

40 votes
40 votes
Best answer
If $x$ is a fly, then for all $y$ which eats $x$, $y$ is a bat. This means only bats eat flies.

Option (E).
edited by
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4 Comments

Sir, isn't option (b) and (e) equivalent to each other? Why option (B) is not the correct choice?
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@Arjun sir, @sourav.

Is this correct for option c

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@Arjun sir But if y eats x then only it will be implied as a bat??so how you can say that not all bats eat fly?please explain
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11 votes
11 votes
For any/all x if it is a fly and if it gets eaten up by any y than that y must be a bat..

So only bats eat flies.

Option e
3 votes
3 votes

first solve ∀y(E(y,x)→B(y)) only take x=fly1

so for x1fly1  ∀y(E(y,x)→B(y)) telling that : (if y1 eats fly1  then y1 is Bat) and(if y2 eats fly1  then y2 is Bat) and (if y1 eats fly1  then y1 is Bat ) and ....(if yn eats fly1  then yn is Bat).

this conclude that:  only bats eats fly1 .

no solve comlete :∀x(F(x)→∀y(E(y,x)→B(y)))

 = (if x is fly1 then only bats eats fly1) and (if x is fly2 then only bats eats fly2) and (if x is fly3 then only bats eats fly3)...

= every fly  eaten by bats only.

= only bats eats flies.

Ans: E

 

Answer:

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