Universal Quantifier

Definition

The universal quantifier says that a statement applies to every object in a domain.

Plain form:

For every x, P(x)

Meaning:

P is true for all x in the relevant domain.

Universal quantifier = all cases.

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Simple Example

Example:

For every x, if x is divisible by 4, then x is even.

This says the rule applies to all relevant values of x.

It is not about one number.

It is about every number in the domain.

Universal statements make claims about all objects under consideration.

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Domain Matters

The domain is the set of objects being discussed.

Example:

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

This is true.

But:

For every x in {2, 3, 4}, x is even.

is false.

The statement changed because the domain changed.

Universal quantification depends on the domain.

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One Counterexample Is Enough

A universal statement is false if there is at least one counterexample.

Statement:

For every x, x is even.

Counterexample:

x = 3

Since 3 is not even, the universal statement is false.

One failed case defeats a universal claim.

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Universal Quantifier vs AND

Over a small finite domain, a universal statement acts like a large AND.

Domain:

{a, b, c}

Statement:

For every x, P(x)

Equivalent finite form:

P(a) AND P(b) AND P(c)

All cases must hold.

Universal quantification generalizes AND across a domain.

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Final Summary

• The universal quantifier means "for every."
• It says a predicate holds for all objects in a domain.
• The domain matters.
• One counterexample makes a universal statement false.
• Over a finite domain, universal quantification behaves like AND across all cases.