I'm new to math and self studying at the moment, and I came upon a situation where mechanically negating has led to something that looked wrong. Here is what i think is correct.
$\neg (\forall x \in R$ s.t. $x < 5, \exists y \in R$ s.t. $y <3, $ s.t. $x < 1)$
$\exists x \in R$ s.t. $x < 5, \forall y \in R$ s.t. $y<3, $ s.t. $ x \geq 1$
At first the instinct is to write the same statement that I wrote with $x \geq 5$ and $y \geq 3$. But it seemed wrong to me since:
For all dogs younger than 2, there's an owner older than 4, and the owner has a pigeon.
The correct negation of the above seems to be: There exists a dog younger than 2, for all owners older than 4, and the owner doesn't have a pigeon.
Have I totally misunderstood this?