Textbook
The assertion $\forall x\in A:P(x)$ says that predicate $P(x)$ is true for every element of domain $A$. The quantifier $\forall$ binds $x$ within its scope. A proof must cover an arbitrary allowed element; one counterexample suffices to refute the assertion.
Negating universality gives $\neg(\forall x\in A:P(x))\Leftrightarrow\exists x\in A:\neg P(x)$. The existential symbol here requires one violator. A universal assertion over an empty domain is true because there are no violating elements.
Order matters for nested quantifiers. In $\forall x\in\mathbb R\;\exists y\in\mathbb R:y>x$, the witness $y=x+1$ depends on the chosen $x$. Requiring one fixed $y$ for all $x$ is false: any proposed $y$ is defeated by $x=y+1$.
An arbitrary element in a proof is not a randomly selected or convenient example. The argument may use only properties guaranteed for every element of the stated domain. If a proof additionally assumes positivity, its domain must ensure that assumption. A statement about every element of a subset does not automatically extend to a larger domain. Restricting the domain does preserve a true universal statement, since restriction cannot create new counterexamples.
Definitions and further reading: Mathematics for Computer Science.
In plain language
“Every” extends a claim to each element of the stated domain: three successful cases do not rule out failure in a fourth. One broken chair refutes a claim about all chairs, while inspecting the first row cannot establish it for the entire hall. A general proof can cover the domain without a literal screwdriver tour. Promising a suitable chair for each person permits different chairs rather than demanding one miracle chair for everyone; the order of choices changes the meaning.
Example
For the illustrative set $A=\{2,4,6\}$, the example tests “every element is greater than 3.” The assertion fails as soon as 2 is inspected; the remaining steps display the entire finite domain.
The Universal Quantifier: A Condition for Every Element
Element 2
Two is a counterexample to the universal assertion.
Step 1 of 3: Element 2
Read all steps
Element 2
Two is a counterexample to the universal assertion.
- State
- 2excluded46
Element 4
Four passes, but cannot erase the existing violation.
- State
- 2excluded4included6
Element 6
The domain is exhausted, but one failure already refuted “every.”
- State
- 2excluded4included6included
Two belongs to the stated domain, so its failure genuinely refutes the assertion. A number outside $A$, even one failing the condition, would not be a counterexample here. Four and six succeeding also show that failure does not mean “nobody qualifies”: some elements pass. “Not every element” differs from “every element fails.” The former requires at least one failure; the latter forbids every success.
Related concepts: Predicates, The existential quantifier, Proof.