What is negation of the following sentence?

Question

What is negation of All birds can fly.

The question seems bit funny but i don't know which of the following two sentences is correct:

1. Some birds can not fly
2. There is at least one bird which can not fly.

Both the sentence seems almost logically same. But which of the following is true.

In book correct option out of four is sentence 1.

Answer 1

$B(x):$ x is a bird.

$F(x):$ x can fly.

All birds can fly: $$\forall x(B(x)\rightarrow F(x))$$

Negation of the above $$\lnot \forall x(B(x)\rightarrow F(x))\equiv \exists x \Big(\lnot\big(\lnot B(x) \lor F(x)\big)\Big) \equiv \exists x (B(x) \land \lnot F(x)$$

You can certainly translate the statement as you did, and your translations are equivalent: mathematically, some is equivalent to at least one. So you may as well stick with "some".

An alternative translation of the negation would be simply the negation symbolized at the very left: "Not all birds can fly."

Answer 2

I would say only 2 is the negation of All birds can fly.

All birds can fly is only true if all birds can do it.

So if not all birds can fly it would be a negation.

There is at least one bird which can not fly, is equal to not all birds can fly

Answer 3

Both the sentence are correct.

I have come through many questions like these in which two answer are correct, when i asked to my teacher about it teacher he told me to mark first correct option.

So if it is asked as multiple choice question then you should mark which ever comes first.

Answer 4

If "some birds" is synonymous with "at least one bird", the two are equivalent. Whether the plural form implies "at least two" rather than "at least one" may be debatable, but this is a question of English rather than mathematics. "There is at least one bird which can not fly" is unambiguous.

Answer 5

Translated in to set-language: "Every element of the set of birds $B$ belongs to the set of animals that fly $F$". So this translates as $B \subset F$. Negating this would be the equivalent of saying $B \nsubseteq F$. So in set language this would translate:"At least one bird cannot fly"(there is an element of $B$ that does not ly in $F$).