Negate ∀n (2n > n) Step by Step | Negating with quantifiers

Step by step

The question

Question: write the opposite of ∀n (2n > n).

∀n (2n > n)

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.

= ¬∀n (2n > n)

2 Move the not inside

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

So ¬∀ becomes ∃¬.

≡ ∃n ¬(2n > n)

¬∀ = ∃¬

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 ∀n ¬(2n > n) ✗

3 Turn the inside around

The opposite of 2n > n is 2n ≤ n.

Every number that is not one is the other.

≡ ∃n (2n ≤ n)

¬(a > b) is a ≤ b

4 Say it in words

In words: there is some n, with 2n ≤ n.

= there is some n: 2n ≤ n

5 Check

Let's check with n = 3.

The first inside: 6 > 3, that is true.

n = 3: 2n > n gives 6 > 3: T

The new inside: 6 ≤ 3, that is false.

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

Right!

n = 3: 2n ≤ n gives 6 ≤ 3: 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