Negate ∀x ∀y (x + y = y + x)

Step by step

The question

Question: write the opposite of ∀x ∀y (x + y = y + 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 ∃¬.

Not true for every y means: there is some y 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 + y = y + x is x + y ≠ y + x.

4 Say it in words

In words: there is some x, there is some y, with x + y ≠ y + x.

5 Check

Let's check with x = 3, y = 1.

The first inside: 4 = 4, that is true.

The new inside: 4 ≠ 4, 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.