Negate ∃x ∃y (x - y = 3) Step by Step | Negating with quantifiers

Step by step

The question

Question: write the opposite of ∃x ∃y (x − y = 3).

∃x ∃y (x − y = 3)

The backwards E means: there is some.

∀ = for every

∃ = 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.

= ¬∃x ∃y (x − y = 3)

2 Move the not inside

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

So ¬∃ becomes ∀¬.

≡ ∀x ¬∃y (x − y = 3)

¬∃ = ∀¬

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

So ¬∃ becomes ∀¬.

≡ ∀x ∀y ¬(x − y = 3)

¬∃ = ∀¬

Careful!

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

Careful: not ∃x ¬(x − y = 3) ✗

3 Turn the inside around

The opposite of x − y = 3 is x − y ≠ 3.

≡ ∀x ∀y (x − y ≠ 3)

¬(a = b) is a ≠ b

4 Say it in words

In words: for every x, for every y, with x − y ≠ 3.

= for every x, for every y: x − y ≠ 3

5 Check

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

The first inside: 2 = 3, that is false.

x = 3, y = 1: x − y = 3 gives 2 = 3: F

The new inside: 2 ≠ 3, that is true.

One true, one false: opposite, as they must be.

Right!

x = 3, y = 1: x − y ≠ 3 gives 2 ≠ 3: T

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.

Answer