Negate ∃x ∀y (x ≤ y)

Step by step

The question

Question: write the opposite of ∃x ∀y (x ≤ y).

The upside-down A means: for every.

The backwards E means: 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.

2 Move the not inside

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

So ¬∃ becomes ∀¬.

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

So ¬∀ becomes ∃¬.

Careful!

The opposite of less than or equal must also take care of equal.

3 Turn the inside around

The opposite of x ≤ y is x > y.

Every number that is not one is the other.

4 Say it in words

In words: for every x, there is some y, with x > y.

5 Check

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

The first inside: 3 ≤ 1, that is false.

The new inside: 3 > 1, that is true.

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.