Quantifiers and negation
Prerequisite: you should already be comfortable reading short logical statements from 1.1 Propositional logic.
What the quantifiers say
Definition
Universal and existential quantifiers
∀x P(x) means that P(x) is true for every allowed value of x.
∃x P(x) means that there is at least one allowed value of x for which
P(x) is true.
The allowed values of x come from the domain. Always keep the domain in mind.
Negating quantifiers
The negation rules are small, but they matter a lot:
| Statement | Negation |
| --- | --- |
| ∀x P(x) | ∃x ¬P(x) |
| ∃x P(x) | ∀x ¬P(x) |
Work through one example
Worked example
Negating a universal statement
Original statement:
For every real number x, x^2 >= 0.
Negation:
There exists a real number x such that x^2 < 0.
Why this works:
- Change "for every" to "there exists".
- Negate the inside statement.
- Keep the same domain.
Solution
The pattern to remember
Common mistake
Common mistake
Do not stop after changing only the quantifier
¬∀x P(x) is not the same as ∀x ¬P(x). The first says "not every x works",
which is the same as "there exists an x for which P(x) fails".
Quick check
Quick check
Write the negation of `∃x (x is a student and x has submitted the form)`
Try it live
Try it here
Quantifier negation stepper
The live stepper reveals one quantifier-negation move at a time.
Example
For every real number x, x^2 >= 0.
- 1. Start with the outer quantifier: “for every x.”