Negate ∀x (x^2 ≥ 0) Step by Step | Negating with quantifiers

Step by step

The question

Question: write the opposite of ∀x (x² ≥ 0).

∀x (x² ≥ 0)

The upside-down A means: for every.

∀ = 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 (x² ≥ 0)

2 Move the not inside

Not true for every x means: there is some x where it fails.

So ¬∀ becomes ∃¬.

≡ ∃x ¬(x² ≥ 0)

¬∀ = ∃¬

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.

Careful: not ∀x ¬(x² ≥ 0) ✗

3 Turn the inside around

The opposite of x² ≥ 0 is x² < 0.

Every number that is not one is the other.

≡ ∃x (x² < 0)

¬(a ≥ b) is a < b

4 Say it in words

In words: there is some x, with x² < 0.

= there is some x: x² < 0

5 Check

Let's check with x = 3.

The first inside: 9 ≥ 0, that is true.

x = 3: x² ≥ 0 gives 9 ≥ 0: T

The new inside: 9 < 0, that is false.

One true, one false: opposite, as they must be.

Right!

x = 3: x² < 0 gives 9 < 0: F

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