Steply

Negating with quantifiers

Discrete Math · 20 problems, easy to harder

▶ Watch the first one solved

How we do it

  1. Put not in front
  2. Move the not inside
  3. Turn the inside around
  4. Say it in words
  5. Check

The steps for Negate ∀x (x^2 ≥ 0). Every solution names its steps, so you always know what comes next.

Worked examples

Negate ∀x (x^2 ≥ 0) Easy

The question

Question: write the opposite of ∀x (x² ≥ 0).

∀x (x2 ≥ 0)

The upside-down A means: for every.

∀ = 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 (x2 ≥ 0)

2 Move the not inside

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

So ¬∀ becomes ∃¬.

≡ ∃x ¬(x2 ≥ 0)
¬∀ = ∃¬

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.

Careful: not ∀x ¬(x² ≥ 0) ✗

3 Turn the inside around

The opposite of x² ≥ 0 is x² < 0.

Every number that is not one is the other.

≡ ∃x (x2 < 0)
¬(a ≥ b) is a < b

4 Say it in words

In words: there is some x, with x² < 0.

= there is some x: x² < 0

5 Check

Let's check with x = 3.

The first inside: 9 ≥ 0, that is true.

x = 3: x2 ≥ 0 gives 9 ≥ 0: T

The new inside: 9 < 0, that is false.

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

Right!

x = 3: x2 < 0 gives 9 < 0: F

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
¬(a ≥ b) is a < b

▶ Watch it solved

Negate ∀x ∃y (x + y = 0) Medium

The question

Question: write the opposite of ∀x ∃y (x + y = 0).

∀x ∃y (x + y = 0)

The upside-down A means: for every.

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 = 0)

2 Move the not inside

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

So ¬∀ becomes ∃¬.

≡ ∃x ¬∃y (x + y = 0)
¬∀ = ∃¬

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

So ¬∃ becomes ∀¬.

≡ ∃x ∀y ¬(x + y = 0)
¬∃ = ∀¬

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.

Careful: not ∀x ¬(x + y = 0) ✗

3 Turn the inside around

The opposite of x + y = 0 is x + y ≠ 0.

≡ ∃x ∀y (x + y ≠ 0)
¬(a = b) is a ≠ b

4 Say it in words

In words: there is some x, for every y, with x + y ≠ 0.

= there is some x, for every y: x + y ≠ 0

5 Check

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

The first inside: 4 = 0, that is false.

x = 3, y = 1: x + y = 0 gives 4 = 0: F

The new inside: 4 ≠ 0, that is true.

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

Right!

x = 3, y = 1: x + y ≠ 0 gives 4 ≠ 0: 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
¬(a = b) is a ≠ b

▶ Watch it solved

Negate ∀x ∀y (x + y = y + x) Harder

The question

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

∀x ∀y (x + y = y + x)

The upside-down A means: for every.

∀ = 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 = y + x)

2 Move the not inside

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

So ¬∀ becomes ∃¬.

≡ ∃x ¬∀y (x + y = y + x)
¬∀ = ∃¬

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

So ¬∀ becomes ∃¬.

≡ ∃x ∃y ¬(x + y = y + x)
¬∀ = ∃¬

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.

Careful: not ∀x ¬(x + y = y + x) ✗

3 Turn the inside around

The opposite of x + y = y + x is x + y ≠ y + x.

≡ ∃x ∃y (x + y ≠ y + x)
¬(a = b) is a ≠ b

4 Say it in words

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

= there is some x, there is some y: x + y ≠ y + x

5 Check

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

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

x = 3, y = 1: x + y = y + x gives 4 = 4: T

The new inside: 4 ≠ 4, that is false.

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

Right!

x = 3, y = 1: x + y ≠ y + x gives 4 ≠ 4: F

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
¬(a = b) is a ≠ b

▶ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. negate ∀x (x^2 ≥ 0)
    Answer
    ¬(a ≥ b) is a < b
  2. negate ∃x (2x + 1 = 7)
    Answer
    ¬(a = b) is a ≠ b
  3. negate for all x, x + 1 > x
    Answer
    ¬(a > b) is a ≤ b
  4. negate there exists x such that x^2 = 4
    Answer
    ¬(a = b) is a ≠ b
  5. negate ∀x (x ≥ 0)
    Answer
    ¬(a ≥ b) is a < b
  6. negate ∃x (x < 3)
    Answer
    ¬(a < b) is a ≥ b
  7. negate ∀n (2n > n)
    Answer
    ¬(a > b) is a ≤ b

Medium

  1. negate ∀x ∃y (x + y = 0)
    Answer
    ¬(a = b) is a ≠ b
  2. negate ∃x ∀y (x ≤ y)
    Answer
    ¬(a ≤ b) is a > b
  3. negate ∀x (x > 2 → x^2 > 4)
    Answer
    ≡ ∃x (x > 2 ∧ x2 ≤ 4)
  4. negate ∃x (x > 0 ∧ x < 1)
    Answer
    ≡ ∀x (x ≤ 0 ∨ x ≥ 1)
  5. negate ∀x (x < 0 ∨ x ≥ 1)
    Answer
    ≡ ∃x (x ≥ 0 ∧ x < 1)
  6. negate ∀x (3x - 1 ≠ 5)
    Answer
    ¬(a ≠ b) is a = b
  7. negate ∃y (y^2 + 1 = 0)
    Answer
    ¬(a = b) is a ≠ b
  8. negate ∃n (n^2 < n)
    Answer
    ¬(a < b) is a ≥ b

Harder

  1. negate ∀x ∀y (x + y = y + x)
    Answer
    ¬(a = b) is a ≠ b
  2. negate ∃x ∃y (x - y = 3)
    Answer
    ¬(a = b) is a ≠ b
  3. negate ∀x ∃y (y > x^2)
    Answer
    ¬(a > b) is a ≤ b
  4. negate ∃x (x^2 = 9 ∧ x < 0)
    Answer
    ≡ ∀x (x2 ≠ 9 ∨ x ≥ 0)
  5. negate ∀x (x^2 > 4 → x > 2)
    Answer
    ≡ ∃x (x2 > 4 ∧ x ≤ 2)