Step by step
The question
Question: write the opposite of ∀x (x < 0 ∨ x ≥ 1).
The upside-down A means: for every.
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 ∃¬.
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
De Morgan: the not goes onto each part, and or turns into and.
Now each part: x < 0 becomes x ≥ 0, and x ≥ 1 becomes x < 1.
4 Say it in words
In words: there is some x, with x ≥ 0 ∧ x < 1.
5 Check
Let's check with x = 3.
The first inside: 3 < 0 ∨ 3 ≥ 1, that is true.
The new inside: 3 ≥ 0 ∧ 3 < 1, that is false.
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.