Negate ∃n (n^2 < n) Step by Step | Negating with quantifiers

Step by step

The question

Question: write the opposite of ∃n (n² < n).

∃n (n² < n)

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.

= ¬∃n (n² < n)

2 Move the not inside

There is no n means: for every n, it fails.

So ¬∃ becomes ∀¬.

≡ ∀n ¬(n² < n)

¬∃ = ∀¬

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 n² < n is n² ≥ n.

Every number that is not one is the other.

≡ ∀n (n² ≥ n)

¬(a < b) is a ≥ b

4 Say it in words

In words: for every n, with n² ≥ n.

= for every n: n² ≥ n

5 Check

Let's check with n = 3.

The first inside: 9 < 3, that is false.

n = 3: n² < n gives 9 < 3: F

The new inside: 9 ≥ 3, that is true.

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

Right!

n = 3: n² ≥ n gives 9 ≥ 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