Negate ∃x (x > 0 ∧ x < 1)

Step by step

The question

Question: write the opposite of ∃x (x > 0 ∧ x < 1).

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 ∀¬.

Careful!

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

3 Turn the inside around

De Morgan: the not goes onto each part, and and turns into or.

Now each part: x > 0 becomes x ≤ 0, and x < 1 becomes x ≥ 1.

4 Say it in words

In words: for every x, with x ≤ 0 ∨ x ≥ 1.

5 Check

Let's check with x = 3.

The first inside: 3 > 0 ∧ 3 < 1, that is false.

The new inside: 3 ≤ 0 ∨ 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.