Step by step
The question
Question: write the opposite of ∀x ∃y (y > x²).
The upside-down A means: for every.
The backwards E means: there is some.
1 Put not in front
The opposite of a statement: we put not in front of all of it.
Now we move the not inside, one piece at a time.
2 Move the not inside
Not true for every x means: there is some x where it fails.
So ¬∀ becomes ∃¬.
There is no y means: for every y, it fails.
So ¬∃ becomes ∀¬.
Careful!
Not every does not mean none.
If not every student is here, some student is missing.
It does not mean every student is missing.
3 Turn the inside around
The opposite of y > x² is y ≤ x².
Every number that is not one is the other.
4 Say it in words
In words: there is some x, for every y, with y ≤ x².
5 Check
Let's check with x = 3, y = 1.
The first inside: 1 > 9, that is false.
The new inside: 1 ≤ 9, that is true.
One true, one false: opposite, as they must be.
Right!
Let's look back.
First: put not in front.
Then: move the not inside.
Then: turn the inside around.
Then: say it in words.
Last: check.