Step by step
The question
Question: write the opposite of ∃x (x < 3).
∃x (x < 3)
The backwards E means: there is some.
∀ = for every
∃ = 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.
= ¬∃x (x < 3)
2 Move the not inside
There is no x means: for every x, it fails.
So ¬∃ becomes ∀¬.
≡ ∀x ¬(x < 3)
¬∃ = ∀¬
Careful!
The opposite of less than must also take care of equal.
Careful: not ¬(a < b) is a > b ✗
3 Turn the inside around
The opposite of x < 3 is x ≥ 3.
Every number that is not one is the other.
≡ ∀x (x ≥ 3)
¬(a < b) is a ≥ b
4 Say it in words
In words: for every x, with x ≥ 3.
= for every x: x ≥ 3
5 Check
Let's check with x = 3.
The first inside: 3 < 3, that is false.
x = 3: x < 3 gives 3 < 3: F
The new inside: 3 ≥ 3, that is true.
One true, one false: opposite, as they must be.
Right!
x = 3: x ≥ 3 gives 3 ≥ 3: T
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.
Answer
¬(a < b) is a ≥ b