Negate ∃y (y^2 + 1 = 0)

Step by step

The question

Question: write the opposite of ∃y (y² + 1 = 0).

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 y means: for every y, it fails.

So ¬∃ becomes ∀¬.

Careful!

There is none means every one fails, not just one.

3 Turn the inside around

The opposite of y² + 1 = 0 is y² + 1 ≠ 0.

4 Say it in words

In words: for every y, with y² + 1 ≠ 0.

5 Check

Let's check with y = 3.

The first inside: 10 = 0, that is false.

The new inside: 10 ≠ 0, 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.