Negation of Quantified Statements

Definition

Negating a quantified statement changes what kind of claim is being made.

Universal statement:

For every x, P(x)

Negation:

There exists x such that NOT P(x)

Existential statement:

There exists x such that P(x)

Negation:

For every x, NOT P(x)

Negating a quantifier switches all to at least one counterexample, and existence to none.

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Negating A Universal Statement

Statement:

For every x, P(x)

This says every case satisfies P.

To deny it, we only need one case that fails.

Negation:

There exists x such that NOT P(x)

Example:

For every x, x is even.

Negation:

There exists x such that x is not even.

The negation of "all" is "at least one not."

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Negating An Existential Statement

Statement:

There exists x such that P(x)

This says at least one case satisfies P.

To deny it, every case must fail.

Negation:

For every x, NOT P(x)

Example:

There exists x such that x is even.

Negation:

For every x, x is not even.

The negation of "exists" is "all not."

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Why Not Just Add NOT?

It is not enough to place NOT casually without changing the quantifier.

Incorrect pattern:

NOT for every x, P(x)

Clear equivalent:

There exists x such that NOT P(x)

The clear version explains what would make the original universal statement false.

Negation should expose the actual counterclaim.

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Domain Still Matters

Negation depends on the same domain.

Statement:

For every x in {2, 4, 6}, x is even.

Negation:

There exists x in {2, 4, 6} such that x is not even.

The domain does not change.

Only the quantifier and predicate are negated.

Keep the same domain when negating quantified statements.

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Final Summary

• Negating a universal statement produces an existential counterexample.
• For every x, P(x) becomes There exists x such that NOT P(x).
• Negating an existential statement produces a universal denial.
• There exists x such that P(x) becomes For every x, NOT P(x).
• The domain stays the same.
• Clear negation shows what would make the original statement false.