Negate for all x, x + 1 > x

Step by step

The question

Question: write the opposite of ∀x (x + 1 > x).

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

The opposite of x + 1 > x is x + 1 ≤ x.

Every number that is not one is the other.

4 Say it in words

In words: there is some x, with x + 1 ≤ x.

5 Check

Let's check with x = 3.

The first inside: 4 > 3, that is true.

The new inside: 4 ≤ 3, 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.